Scattering threshold for the focusing nonlinear Klein–Gordon equation

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SCATTERING THRESHOLD FOR

THE FOCUSING NONLINEAR KLEIN-GORDON EQUATION

S. IBRAHIM, N. MASMOUDI, AND K. NAKANISHI

Abstract. We show scattering versus blow-up dichotomy below the ground stateenergy for the focusing nonlinear Klein-Gordon equation, in the spirit of Kenig-Merle for the H1 critical wave and Schrodinger equations. Our result includes theH1 critical case, where the threshold is given by the ground state for the masslessequation, and the 2D square-exponential case, where the mass for the groundstate may be modified, depending on the constant in the sharp Trudinger-Moserinequality. The main difficulty is the lack of scaling invariance in both the linearand the nonlinear terms.

Contents

1. Introduction 21.1. The problem and overview 21.2. Main result 41.3. Some notation 92. Variational characterizations 92.1. Energy landscape in various scales 102.2. Ground state as common minimizer 132.3. H1 critical case; massless threshold 142.4. Exponential case; mass-modified threshold 152.5. Parameter independence of the splitting 192.6. Variational estimates 203. Blow-up 224. Global space-time norm 224.1. Reduction to the first order equation 234.2. Strichartz-type estimates and exponents 244.3. Global perturbation of Strichartz norms 265. Profile decomposition 335.1. Linear profile decomposition 335.2. Nonlinear profile decomposition 396. Extraction of a critical element 457. Extinction of the critical element 477.1. Compactness 477.2. Zero momentum and non-propagation 487.3. Dispersion and contradiction 50Appendix A. The range of scaling exponents 52

2010 Mathematics Subject Classification. 35L70, 35B40, 35B44, 47J30.Key words and phrases. nonlinear Klein-Gordon equation, scattering theory, blow-up solution,

ground state, Sobolev critical exponent, Trudinger-Moser inequality.1

http://arxiv.org/abs/1001.1474v3

2 S. IBRAHIM, N. MASMOUDI, AND K. NAKANISHI

Appendix B. Table of Notation 53Acknowledgments 54References 54

1. Introduction

1.1. The problem and overview. We study global and asymptotic behavior ofsolutions in the energy space for the nonlinear Klein-Gordon equation (NLKG):

u−∆u+ u = f ′(u), u : R1+d → R, (d ∈ N) (1.1)

where f : R → R is a given function. Typical examples that we can treat are thepower nonlinearities in any dimensions

f(u) = λ|u|p+2, (2⋆ < p+ 2 ≤ 2⋆, λ ≥ 0), (1.2)

where 2⋆ and 2⋆ respectively denote the L2 and H1 critical powers

2⋆ = 2 +4

d, 2⋆ =

{2 + 4

d−2(d ≥ 3)

∞ (d ≤ 2), (1.3)

and the square-exponential nonlinearity in two spatial dimensions

f(u) = λ|u|peκ|u|2, (d = 2, p > 4, λ ≥ 0, κ > 0), (1.4)

which is related to the critical case for the Trudinger-Moser inequality. The equationconserves (at least formally) the energy

E(u; t) = E(u(t), u(t)) :=

∫

Rd

|u|2 + |∇u|2 + |u|22

− f(u)dx. (1.5)

The main goal in this paper is to give necessary and sufficient conditions for thesolution u to scatter, which means that u is asymptotic to some free solutions as t→±∞, under the condition that u has less energy than the least energy static solution,namely the ground state. In the defocusing case, where f has the opposite sign, onehas the scattering result for all finite energy solutions, see [9, 17, 36, 35, 37, 23].In the focusing case, it turns out that the solutions below the ground energy splitinto the scattering solutions and the blow-up solutions (in both time directions inboth cases). Such results have been recently established for many other equationsincluding the nonlinear wave equation (NLW), the nonlinear Schrodinger equation(NLS), the Yang-Mills system and the wave maps, since Kenig-Merle’s [27] on NLSwith theH1 critical power (i.e. p+2 = 2⋆ in (1.2)), see [4, 13, 15, 26, 28, 29, 30, 42, 45]and the references therein.

To be more precise, let us recall the result by Kenig-Merle for the critical nonlinearwave equation

u−∆u = f ′(u), f(u) = |u|2⋆. (1.6)

Let E(0)(u) be the conserved energy, and Q be a static solution with the least energy:

E(0)(u) :=

∫

Rd

|u|2 + |∇u|22

+ f(u)dx, Q(x) :=

[1 +

|x|2d(d− 2)

]−(d−2)/2

. (1.7)

SCATTERING THRESHOLD FOR NLKG 3

Kenig-Merle [26] proved that every solution with E(0)(u) < E(0)(Q) scatters in theenergy space as t → ±∞, provided that ‖∇u(0)‖L2 < ‖∇Q‖L2 , and otherwise itblows up in finite time both for t > 0 and for t < 0. The idea of their proof isto bring the concentration compactness argument into the scattering problem byusing space-time norms and the concept of “critical element”, that is the minimalnon-scattering solution.

The equations in those papers following Kenig-Merle have a common importantproperty—the scaling invariance. It is further shared with the solution space (eitherthe energy space or L2, i.e. the critical case), except for the NLS with a subcriticalpower [15, 4]. The scaling invariance brings significant difficulties for the analysis,but also a lot of algebraic or geometric structures and simplifications. Hence it isa natural question what happens if the invariance is broken in the linear and thenonlinear parts of the equation. This is the main technical challenge in this paper.

The dichotomy into the global existence and the blow-up has been known [39]long before the scattering result of Kenig-Merle, under the name of “potential well”,which is defined by derivatives of the static energy functional. More precisely, Payne-Sattinger [39] proved on bounded domains the dichotomy into blow-up and globalexistence for solutions below the ground energy, by the sign of the functional

K1,0(u) :=

∫|∇u|2 + |u|2 − uf ′(u)dx. (1.8)

It is easy to observe that their argument applies to the whole space Rd as soon as onehas the local wellposedness in the energy space. Hence our primary task is to provethe scattering result in the region of global existence. Then our first problem due tothe inhomogeneity is that the above functional K1,0 is not suited for the scatteringproof, though it is useful for the blow-up and global existence. More specifically, wewant to use the functional

Kd,−2(u) :=

∫2|∇u|2 + d[uf ′(u)− 2f(u)]dx, (1.9)

which is related to the virial identity. There is actually a one-parameter family offunctionals, corresponding to various scalings, each of which defines a splitting ofthe solutions below the ground energy by its sign. For example, Shatah [41] usedanother functional

K0,1(u) :=

∫d− 2

2|∇u|2 + d

2|u|2 − df(u)dx, (1.10)

to prove the instability of the standing waves. Note that in his proof the instabilityis not given by blow-up in the region K0,1(u) < 0. More recently, Ohta-Todorova[38] proved blow-up in the region Kd,−2(u) < 0, but they need radial symmetry forthe powers p close to 2⋆.

The special feature of the critical wave equation (1.6) is that those functionals arethe same modulo constant multiples, which is exactly due to the scaling invariance.For the NLS with a subcritical power [15, 4], the functionals are different from eachother, but the situation is much better than NLKG, because they contain only twoterms (without the L2 norm), the L2 is another conserved quantity, and the virialidentity is used both for the blow-up and for the scattering, while K1,0 is not souseful for NLS.


It turns out, however, that those algebraically different functionals for NLKGdefine the same splitting below the threshold energy. This observation does notseem to be well recognized, but it is indeed crucial for the proof of the dichotomy,since we need different functionals for the blow-up and for the scattering.

One interesting feature resulting from the breakdown of the scaling is that, forsome nonlinearity, the energy threshold is not given by the ground state of theoriginal NLKG, but by that of a modified equation. More precisely, for the H1

critical power (p + 2 = 2⋆) in three dimensions or higher, the threshold is givenby that of the critical wave equation, or massless Klein-Gordon equation. This canbe expected because the concentration by the critical scaling makes the L2 normvanish while preserving other components, namely the massless energy. However thetransition from the Klein-Gordon to the wave requires non-trivial amount of effortin the scattering proof.

We find another instance of mass modification, which is more surprising. That isin two dimensions and for nonlinearities which grow slightly slower than the squareexponential e|u|

2, where the mass for the threshold ground energy can change to any

number between 0 and 1, depending on the constant in the sharp (L2) Trudinger-Moser inequality. Thus we prove the existence of extremizers as well as the groundstates with mass less than or equal to the sharp constant, which also seems newfor general nonlinearity on the whole plane. For the existence of the ground stateon bounded domains, we refer to [14, 2, 3]. One should be warned, however, thatthe situation on the whole plane is different from that on disks, unlike the higherdimensional Sobolev critical case, since here the concentration compactness has tobe accompanied with a leak of L2 norm to the spatial infinity. This will be discussedseparately in a forthcoming paper [24].

It is worth noting that the scattering result in the focusing exponential case is ac-tually easier to obtain than in the defocusing case, concerning the global Strichartzestimate. This is because the (mass-modified) ground energy threshold implies thatour solutions are in the subcritical regime for the Trudinger-Moser inequality. Henceconcentration of energy is a priori precluded, and so we do not need the concentra-tion radius or the localized Strichartz estimate used in [23] on the Trudinger-Moserthreshold in the defocusing case. This is another striking difference from the powercase, where the analysis for the focusing case essentially contains that for the defo-cusing case.

1.2. Main result. To state the main results of this paper, we need to introducesome notation and assumptions for the variational setting and the nonlinear settingof the problem.

1.2.1. Variational setting. To specify our class of solutions, we need the static energy

J(ϕ) :=1

2

∫

Rd

[|∇ϕ|2 + |ϕ|2]dx− F (ϕ), F (ϕ) :=

∫

Rd

f(ϕ)dx, (1.11)

and its derivatives with respect to different scalings. In the critical/exponentialcases, we also need the energy with a modified mass c ≥ 0,

J (c)(ϕ) =1

2

∫

Rd

[|∇ϕ|2 + c|ϕ|2]dx− F (ϕ). (1.12)


For any α, β, λ ∈ R and ϕ : Rd → R, we define the two-parameter rescaling family

ϕλα,β(x) = eαλϕ(e−βλx), (1.13)

and the differential operator Lα,β acting on any functional S : H1(Rd) → R by

Lα,βS(ϕ) =d

dλ

∣∣∣∣λ=0

S(ϕλα,β). (1.14)

The scaling derivative of the static energy is denoted by

Kα,β(ϕ) := Lα,βJ(ϕ)

=

∫

Rd

[2α+ (d− 2)β

2|∇ϕ|2 + 2α+ dβ

2|ϕ|2 − αϕf ′(ϕ)− dβf(ϕ)

]dx,

K(c)α,β(ϕ) := Lα,βJ (c)(ϕ).

(1.15)

For each (α, β) ∈ R2 in the range

α ≥ 0, 2α+ dβ ≥ 0, 2α+ (d− 2)β ≥ 0, (α, β) 6= (0, 0), (1.16)

we consider the constrained minimization problem

mα,β = inf{J(ϕ) | ϕ ∈ H1(Rd), ϕ 6= 0, Kα,β(ϕ) = 0}. (1.17)

We will prove that it is attained, (after a modification of the mass in some cases),provided that (α, β) is in the above range (1.16). The condition on (α, β) is alsonecessary in general (see Proposition A.1).

Our solutions start from the following subsets of the energy space

K+α,β = {(u0, u1) ∈ H1(Rd)× L2(Rd) | E(u0, u1) < mα,β , Kα,β(u0) ≥ 0},

K−α,β = {(u0, u1) ∈ H1(Rd)× L2(Rd) | E(u0, u1) < mα,β , Kα,β(u0) < 0}.

(1.18)

1.2.2. Nonlinear setting. For the nonlinearity f , we consider the following threecases: the H1 subcritical (d ≥ 1), the 2D exponential case, and the H1 critical(d ≥ 3) cases. First we assume that f : R → R is C2 and

f(0) = f ′(0) = f ′′(0) = 0. (1.19)

Secondly for the variational arguments, we need some monotonicity and convexityconditions. Let D denote the linear operator defined by

Df(u) := uf ′(u). (1.20)

We assume that f satisfies for some ε > 0,

(D − 2⋆ − ε)f ≥ 0, (D − 2)(D − 2⋆ − ε)f ≥ 0, (1.21)

which implies in particular that

D2f ≥ (2⋆ + ε)Df ≥ (2⋆ + ε)2f ≥ 0. (1.22)

Finally we need regularity and growth conditions, which can differ for small |u|and large |u|. Fix a cut-off function χ ∈ C∞

0 (R) satisfying χ(r) = 1 for |r| ≤ 1 andχ(r) = 0 for |r| ≥ 2, and denote

χR(x) := χ(|x|/R), (1.23)


for arbitrary vector x and R > 0. Decompose the nonlinearity by

fS(u) := χ1(u)f(u), fL(u) = f(u)− fS(u). (1.24)

We assume that for some p1 > 2⋆ − 2{|f ′′S(u)|. |u|p1 (d ≤ 4),

|f ′′S(u1)− f ′′

S(u2)|. |u1 − u2|p1 (d ≥ 5),(1.25)

where we should choose p1 < 1 for d ≥ 5.For the behavior of f for large |u|, we distinguish three cases:(1) H1 subcritical case: We assume that for some p2 < 2⋆ − 2

|f ′′L(u)|. |u|p2 (2 ≤ d ≤ 4)

|f ′′L(u1)− f ′′

L(u2)|. (|u1|+ |u2|)p2−1|u1 − u2| (d ≥ 5 and p2 ≥ 1)

|f ′′L(u1)− f ′′

L(u2)|. |u1 − u2|p2 (d ≥ 5 and p2 < 1).

(1.26)

p2 = 2⋆ − 2 will be allowed in some of the later arguments. There is no growthrestriction for d = 1. A typical example is

f(u) = λ1|u|q1 + · · ·λk|u|qk, (1.27)

where λj > 0 and 2⋆ < qj < 2⋆ for all j, which satisfies (1.26) as well as (1.19),(1.21) and (1.25).

(2) H1 critical case. We assume

d ≥ 3, f(u) = |u|2⋆/2⋆. (1.28)

In this case, we do not include lower powers in order to avoid their nontrivial effectsin the variational characterization. The absence of lower powers will only be used insection 2. In particular the Strichartz spaces we use in section 4 can handle the sumof a critical power with a subcritical function. For the variational characterization,the case where lower powers are included will be treated in a forthcoming work.

(3) 2D exponential case: We assume that

d = 2, ∃κ0 ≥ 0, s.t.

{∀κ > κ0, lim|u|→∞ f ′′

L(u)e−κ|u|2 = 0,

∀κ < κ0, lim|u|→∞ fL(u)e−κ|u|2 = ∞,

and if κ0 > 0 then lim|u|→∞

fL(u)/DfL(u) = 0.

(1.29)

Then we define C⋆TM

by

C⋆TM

(F ) = sup{2F (ϕ)‖ϕ‖−2L2(R2) | 0 6= ϕ ∈ H1(R2), κ0‖∇ϕ‖2L2(R2) ≤ 4π}. (1.30)

For example, all the conditions are satisfied by

f(u) = eκ0|u|2 − 1− κ0|u|2 −

κ202|u|4 (1.31)

and by

f(u) = |u|peκ0|u|2+γ|u|, (1.32)


where p > 4, κ0 ≥ 0, and max(−γ, 0) ≪ 1 (depending on κ0(p− 4)). More specifi-cally, it suffices to have for all u ∈ [0,∞) that

8κ0u2 + 3γu+ 2(p− 4) > 0, (1.33)

since, putting g := Df/f = 2κ0u2 + γu+ p, we have 2⋆ = 4 and

(D − 2)(D − 4)f = [(g − 4)2 +Dg + 2(g − 4)]f,

Dg + 2(g − 4) = 8κ0u2 + 3γu+ 2(p− 4) = 2[g(3u/2)− 4]− u2/2.

(1.34)

In addition, one can easily observe that C⋆TM(F ) = ∞ if γ ≥ 0 and C⋆

TM(F ) < ∞if γ < 0, using Moser’s sequence of functions for the former, and by the sphericalsymmetrization for the latter (cf. [32, 1, 40]).1

In short, our assumption on f is that

(1.19), (1.21), (1.25), and [(1.26) or (1.28) or (1.29)]. (1.36)

Then by Sobolev or Trudinger-Moser, we observe that F , Lα,βF and L2α,βF are

continuous functionals on H1(Rd).Now we can state our main result. Denote the quadratic part of the energy (i.e.

the linear energy) by

EQ(u; t) = EQ(u(t), u(t)) :=

∫

Rd

|u|2 + |∇u|2 + |u|22

dx. (1.37)

Theorem 1.1. Assume (1.36) for f . Then for all (α, β) in (1.16), both mα,β and

K±α,β are independent of (α, β). Moreover (1.1) is locally wellposed in the energy

space H1 × L2, and

(1) If (u(0), u(0)) ∈ K−α,β, then u extends neither for t→ ∞ nor for t→ −∞ as

the unique strong solution in H1 × L2.

(2) If (u(0), u(0)) ∈ K+α,β, then u scatters both in t → ±∞ in the energy space.

In other words, u is a global solution and there are v± satisfying

v± −∆v± + v± = 0, EQ(u− v±, u− v±) → 0 (t→ ±∞). (1.38)

The dichotomy of global existence versus blow-up in the subcritical case wasessentially given by Payne-Sattinger [39], using K1,0, on bounded domains. Henceour main contribution is the scattering part, and the parameter independence ofK±α,β. The corresponding result in the defocusing case (hence only the scattering)

has been shown by [9, 18] for the subcritical f in three dimensions and higher, by [35]in lower dimensions, by [36] for the H1 critical f , and by [23] for the 2D exponentialnonlinearity. The massless H1 critical case (the other powers cannot be controlledby the massless energy) was solved by [7, 6] for the defocusing f and by [26] for thefocusing nonlinearity.

The parameter independence of mα,β seems to be known in the study of stabilityof standing waves, but the authors could not find an available result as general as

1 Actually, the optimal (fastest) growth to have C⋆

TM(F ) <∞ is given by

f(u) ∼ eκ0|u|2

/|u|2 (|u| → ∞), (1.35)

which will be shown in a forthcoming paper [24]. The results in this paper do not rely on thisobservation, though it seems to have its own interest.


the above one. See [38, 48] for partial results. We quote a recent paper [25] for apure power nonlinearity, but unfortunately their range of (α, β) was not correct (thecondition α ≥ 0 was overlooked; its necessity is shown by Proposition A.1).

The parameter independence of K±α,β, on the other hand, does not seem to have

got much attention from the stability analysis, but it is essential in our proof of thescattering, since the monotonicity is given for the blow-up and for the scattering interms of different Kα,β, respectively K1,0 and Kd,−2.

Thanks to the parameter independence, we may write m = mα,β and K± = K±α,β.

We will also show the following important properties of the energy threshold.

Proposition 1.2. Under the assumptions of the above theorem,

(1) In the subcritical case (1.26), the threshold energy m is attained by some

Q ∈ H1(Rd), independent of (α, β), solving the static equation

−∆Q +Q = f ′(Q), (1.39)

with the least energy J(Q) = m among the solutions in H1(Rd). In other

words, m is attained by the ground states.

(2) In the critical case (1.28), there is no minimizer for (1.17), but we have

m = J (0)(Q), (1.40)

for a static solution Q ∈ H1(Rd) of the massless equation

−∆Q = f ′(Q), (1.41)

with the least massless energy J (0). In other words, m equals to the massless

ground energy.

(3) In the exponential case (1.29), let c := min(1, C⋆TM

(F )), where C⋆TM

(F ) is as

in (1.30). Then we have

m = J (c)(Q), (1.42)

for a static solution Q ∈ H1(R2) of the mass-modified equation

−∆Q + cQ = f ′(Q), (1.43)

with the least energy J (c)(Q). Moreover we have

m ≤ 2π/κ0, (1.44)

where the equality holds if and only if C⋆TM

(F ) ≤ 1, and m = mα,β is attained

in (1.17) if and only if C⋆TM

(F ) ≥ 1.

Again this is well known in the subcritical case. Hence the main novelty is in themass change in the critical/exponential cases. Note that the ground state Q with adifferent mass c ∈ [0, 1) yields standing wave solutions e±itωQ(x) with 1 − ω2 = c.But it is not a true obstruction for the scattering, because its dynamical energy isabove m, although m is the right threshold in the sense that for higher energy levelE > m the sets K± are no longer separated from each other, that is, ∂K+∩∂K− 6= ∅.


1.3. Some notation. Here we recall some standard notation. F denotes the Fouriertransform on Rd, and

〈∇〉 :=√1−∆ = F−1

√1 + |ξ|2F . (1.45)

Lp, Hs, Bsp,q and B

sp,q respectively denote the Lebesgue, Sobolev, inhomogeneous and

homogeneous Besov spaces on Rd. For later use we recall the most used functionals

Kα,β and Hα,β:

K1,0(ϕ) =

∫

Rd

[|∇ϕ|2 + |ϕ|2 − ϕf ′(ϕ)

]dx,

K0,1(ϕ) =

∫

Rd

[d− 2

2|∇ϕ|2 + d

2|ϕ|2 − df(ϕ)

]dx,

Kd,−2(ϕ) =

∫

Rd

[2|∇ϕ|2 − d(D − 2)f(ϕ)

]dx,

(1.46)

H1,0(ϕ) =1

2

∫

Rd

[(D − 2)f(ϕ)] dx,

H0,1(ϕ) =

∫

Rd

[1

d|∇ϕ|2

]dx,

Hd,−2(ϕ) =

∫

Rd

[1

2|ϕ|2 + d

4(D − 2∗)f(ϕ)

]dx.

(1.47)

We give a table of notation in Appendix B.

2. Variational characterizations

In this section, we prove Proposition 1.2. In particular we prove the existence ofground states as constrained minimizers, the (α, β)-independence of the splittings,together with various estimates for solutions below the threshold by variationalarguments, which will be used for the scattering and blowup.

Throughout this section, we assume that (α, β) is in the range (1.16). For easeof presentation, we often omit (α, β) from the subscript. We associate with it thefollowing two numbers:

µ = max(2α + dβ, 2α+ (d− 2)β), µ = min(2α+ dβ, 2α+ (d− 2)β), (2.1)

which come from the scaling exponents for H1 and L2 in (1.13). Notice that in therange (1.16), we have µ > 0, µ ≥ 0, and that α = µ = 0 if and only if (d, α) = (2, 0),which will often be an exceptional case in the following arguments.

We decompose Kα,β = Lα,βJ into the quadratic and the nonlinear parts:

Kα,β = KQα,β +KN

α,β, KQα,β(ϕ) = Lα,β‖ϕ‖2H1/2, KN

α,β(ϕ) = −Lα,βF (ϕ). (2.2)

Then KQα,β(ϕ

λα,β) is non-negative and non-decreasing with respect to λ ∈ R, and

limλ→−∞

KQα,β(ϕ

λα,β) = 0, (2.3)

from its explicit form.


2.1. Energy landscape in various scales. First we investigate how J and itsderivatives behave with respect to the scaling ϕλα,β, in order to getmα,β as a minimaxvalue. The results of this subsection are essentially known, at least under morerestrictions on the nonlinearity and (α, β).

We start from the origin of the energy space.

Lemma 2.1 (Positivity of K near 0). Assume that f satisfies (1.36), and that (α, β)satisfies (1.16) and (d, α) 6= (2, 0). Then for any bounded sequence ϕn ∈ H1(Rd)\{0}such that KQ

α,β(ϕn) → 0, we have for large n,

Kα,β(ϕn) > 0. (2.4)

Note that if (d, α) = (2, 0) then the conclusion is false, since in that caseKQ(ϕλ) =edβλKQ(ϕ) → 0 as λ→ −∞, but K(ϕλ) = edβλK(ϕ) can be negative.

Proof. First we consider the H1 subcritical/critical cases. If d ≥ 2 then

|Df(ϕ)|+ |f(ϕ)|. |ϕ|p1+2 + |ϕ|p2+2, (2.5)

for some 2⋆ < p1 + 2 < p2 + 2 ≤ 2⋆, hence by the Gagliardo-Nirenberg inequality

‖ϕ‖qLqx. ‖∇ϕ‖d(q/2−1)

L2x

‖ϕ‖d−q(d−2)/2L2x

, (2 ≤ q ≤ 2⋆) (2.6)

we obtain

|F (ϕ)|+ |LF (ϕ)|.∑

q=p1+2, p2+2

‖∇ϕ‖d(q/2−1)

L2x

‖ϕ‖d−q(d−2)/2

L2x

. (2.7)

If d = 1 then we can dispose of fL by Sobolev H1(R) ⊂ L∞(R). Then we get

|F (ϕ)|+ |LF (ϕ)|. ‖∇ϕ‖p1/2+1

L2x

‖ϕ‖p1/2+1

L2x

C(‖ϕ‖H1), (2.8)

for some function C determined by fL.Hence if 2α + (d− 2)β > 0 then for any d we have

|KN(ϕ)| = o(‖∇ϕ‖2L2x) = o(KQ(ϕ)). (2.9)

Under the assumption, 2α+(d−2)β = 0 is possible only for d = 1, then using (2.8),

|KN(ϕ)| = o(‖ϕ‖2L2x) = o(KQ(ϕ)). (2.10)

Finally we consider the 2D exponential case (1.29). Then we have

|Df(ϕ)|+ |f(ϕ)|. |ϕ|p(eκ|ϕ|2 − 1), (2.11)

for some p > 2 and any κ > κ0. Since α > 0, we have KQ(ϕn)& ‖∇ϕn‖2L2 → 0, soit suffices to consider ϕ ∈ H1 satisfying for some q > 1 satisfying (4− p)q < 2,

qκ‖∇ϕ‖2L2 ≤ 2π. (2.12)

Let q′ = q/(q − 1) be the Holder conjugate. Then by Holder, Gagliardo-Nirenberg(2.6) and the Trudinger-Moser inequality:

‖∇ϕ‖L2(R2) <√4π =⇒

∫

R2

(e|ϕ|2 − 1)dx.

‖ϕ‖2L2(R2)

4π − ‖∇ϕ‖2L2(R2)

, (2.13)


we obtain

|LF (ϕ)|+ |F (ϕ)|. ‖ϕ‖pLpq′‖eqκ|ϕ|

2 − 1‖1/qL1 ,

. ‖ϕ‖2/q′L2 ‖∇ϕ‖p−2/q′

L2

[ ‖ϕ‖2L2

4π − qκ‖∇ϕ‖2L2

]1/q

. ‖ϕ‖2L2‖∇ϕ‖p−2/q′

L2 .

(2.14)

Since p− 2/q′ > 2 by the choice of q, we get

|KN(ϕ)| = o(‖∇ϕ‖2L2) = o(KQ(ϕ)). (2.15)

Thus in all cases K(ϕ) ∼ KQ(ϕ) > 0 when 0 < KQ(ϕ) ≪ 1. �

The following inequalities describe the graph of J , and will play the central rolein the succeeding arguments.

Lemma 2.2 (Mountain-pass structure). Assume that f satisfies (1.36) and (α, β)satisfies (1.16). Then for any ϕ ∈ H1(Rd) we have

(Lα,β − µ)‖ϕ‖2H1 ≤ −2|β|min(‖ϕ‖2L2, ‖∇ϕ‖2L2),

(Lα,β − µ)F (ϕ) ≥ αεF (ϕ),(2.16)

where ε > 0 is given in (1.21). Hence

(µ− Lα,β)J(ϕ) ≥ αεF (ϕ) + |β|min(‖ϕ‖2L2, ‖∇ϕ‖2L2). (2.17)

Moreover we have

−(Lα,β − µ)(Lα,β − µ)J(ϕ) = (Lα,β − µ)(Lα,β − µ)F (ϕ)

≥ 2αε

d+ 1Lα,βF (ϕ) ≥

2αεµ

d+ 1F (ϕ).

(2.18)

Proof. First we observe that

(L − 2α− (d− 2)β)‖∇ϕ‖2L2x= 0, (L − 2α− dβ)‖ϕ‖2L2

x= 0, (2.19)

and for any functional S of the form S(ϕ) =∫Rd s(ϕ)dx,

Lα,βS(ϕ) =∫

Rd

[(αD + βd)s](ϕ)dx, (2.20)

where Df(ϕ) = ϕf ′(ϕ) as defined in (1.20). Using them, we obtain

(L− µ)‖ϕ‖2H1 = −2|β| ×{‖∇ϕ‖2L2 (β ≥ 0)

‖ϕ‖2L2 (β ≤ 0), (2.21)

and also

LF (ϕ) =∫

[α(D − 2) + 2α+ dβ]f(ϕ)dx

=

∫[(αD − 2α+ 2β) + 2α+ (d− 2)β]f(ϕ)dx.

(2.22)

Since

αD − 2α + 2β = α(D − 2⋆) +2

d(2α+ dβ), (2.23)


using (1.21), we obtain

LF ≥ (µ+ αε)F. (2.24)

Using the above computations, we have

−(L − µ)(L − µ)J(ϕ) = (L − µ)(L − µ)F (ϕ)

= α

∫(αD − 2α + 2β)(D − 2)f(ϕ)dx

≥ αε

∫ [α(D − 2) +

2

d(2α + dβ)

]f(ϕ)dx

≥ 2

d+ 1αεLF (ϕ) ≥ 2αεµ

d+ 1F (ϕ),

(2.25)

where we used (2.23) and (1.21) in the first inequality, min(1, 2/d) ≥ 2/(d + 1) inthe second, and (2.24) in the last. �

Using the above inequalities, we can replace the minimized quantity in (1.17) witha positive definite one, while extending the minimizing region from “the mountainridge” to “the mountain flank”. Let

Hα,β := (1− Lα,β/µ)J. (2.26)

Then the above lemma implies that Hα,β > 0 and

Lα,βHα,β = −(L − µ)(L − µ)J/µ+ µ(1− L/µ)J

≥ 2αε

d+ 1F + µHα,β ≥ 0.

(2.27)

We can rewrite the minimization problem (1.17) by using H :

Lemma 2.3 (Minimization ofH). Assume that f satisfies (1.36) and (α, β) satisfies(1.16). Then mα,β in (1.17) equals

mα,β = inf{Hα,β(ϕ) | ϕ ∈ H1(Rd), ϕ 6= 0, Kα,β(ϕ) ≤ 0}. (2.28)

Proof. Let m′ denote the right hand side of (2.28). Then m ≥ m′ is trivial becauseJ = H if K = 0, so it suffices to show m ≤ m′. Take ϕ ∈ H1 such that K(ϕ) < 0.

If (d, α) 6= (2, 0), then from Lemma 2.1 together with (2.3), we deduce that

(d, α) 6= (2, 0), K(ϕ) < 0 =⇒ ∃λ < 0, K(ϕλ) = 0, H(ϕλ) ≤ H(ϕ), (2.29)

where the latter inequality follows from (2.27) since H(ϕλ) is nondecreasing in λ.Hence m ≤ m′.

If (d, α) = (2, 0), then we use another scaling νu with ν ∈ (0, 1). We haveKQ(νϕ) = ν2KQ(ϕ) and |KN(νϕ)| = o(ν4) by (2.7) or (2.14). Hence K(νϕ) > 0for small ν > 0, and so we deduce

(d, α) = (2, 0), K(ϕ) < 0 =⇒ ∃ν ∈ (0, 1), K(νϕ) = 0, H(νϕ) ≤ H(ϕ), (2.30)

where the inequality follows from H(ϕ) = ‖∇ϕ‖2L2x/2 in this case. Hence m ≤ m′.

�


2.2. Ground state as common minimizer. Now we can prove the parameterindependence ofmα,β via its characterization by the ground states. First we considerthe H1 subcritical case.

Lemma 2.4 (Ground state in the subcritical case). Assume that f satisfies (1.36)and (1.26), and that (α, β) satisfies (1.16). Then mα,β in (1.17) is positive and

independent of (α, β). Moreover mα,β = J(Q) for some Q ∈ H1(Rd) solving the

static NLKG (1.39) with the minimal J(Q) among the solutions in H1(Rd).

Proof. Let ϕn ∈ H1 be a minimizing sequence for (2.28), namely K(ϕn) ≤ 0, ϕn 6= 0and H(ϕn) ց m.

First we consider the case (d, α) 6= (2, 0). Let ϕ∗n be the Schwartz symmetrization

of ϕn, i.e. the radial decreasing rearrangement. Since the symmetrization preservesthe nonlinear parts and does not increase the H1 part, we have ϕ∗

n 6= 0, K(ϕ∗n) ≤

K(ϕn) ≤ 0 and H(ϕn) ≥ H(ϕ∗n) → m. Then using (2.29), we may replace it by

symmetric ψn ∈ H1 such that

ψn 6= 0, K(ψn) = 0, J(ψn) = H(ψn) → m. (2.31)

If α > 0, then (2.17) implies

(µ+ αε)J(ψn) ≥ αε‖ψn‖2H1/2, (2.32)

hence ψn is bounded in H1.If α = 0 (and d > 2), then H(ψn) = ‖∇ψn‖2L2

x/d is bounded, and if ‖ψn‖L2 → ∞,

then by (2.7)

dβ‖ψn‖2L2 ≤ 2KQ(ψn) = −2KN(ψn) ≤ o(‖ψn‖d−2⋆(d−2)/2

L2 ), (2.33)

but since d− 2⋆(d− 2)/2 < 2, this is a contradiction. Hence ψn is bounded in H1.Since ψn is bounded in H1, after replacing with some appropriate subsequence,

it converges to some ψ weakly in H1. By the radial symmetry, it also convergesstrongly in Lp for all 2 < p < 2⋆. Then in the subcritical case (1.26), the nonlinearparts converge, and so K(ψ) ≤ 0 and H(ψ) ≤ m.

If ψ = 0, then K(ψn) = 0 implies that KQ(ψn) = −KN (ψn) → 0, and by Lemma2.1 we have K(ψn) > 0 for large n, a contradiction. Hence ψ 6= 0.

By (2.29), we may replace ψ by its rescaling, so thatK(ψ) = 0, J(ψ) = H(ψ) ≤ mand ψ 6= 0. Then ψ is a minimizer and m = H(ψ) > 0.

Since ψ is a minimizer for (1.17), there is a Lagrange multiplier η ∈ R such that

J ′(ψ) = ηK ′(ψ). (2.34)

Then denoting Lψ = ∂λψλα,β|λ=0, we get

0 = K(ψ) = LJ(ψ) = 〈J ′(ψ)|Lψ〉 = η〈K ′(ψ)|Lψ〉 = ηL2J(ψ). (2.35)

By (2.18) and LJ(ψ) = 0, we have

L2J(ψ) ≤ −µµJ(ψ)− 2αεµ

d+ 1F (ψ) < 0, (2.36)

since µ > 0 or α > 0.


Therefore η = 0 and ψ is a solution to (1.39). The minimality of J(ψ) amongthe solutions is clear from (1.17), since every solution Q in H1 of (1.39) satisfiesK(Q) = 〈J ′(Q)|LQ〉 = 0. This implies that mα,β is independent of (α, β).

In the exceptional case (d, α) = (2, 0), the above argument needs considerablemodifications, due to the scaling invariance

H(ϕ) = ‖∇ϕ‖2L2/2 = H(ϕλ), K(ϕλ) = edβλK(ϕ). (2.37)

First, we should use (2.30) instead of (2.29) to get ψn satisfying (2.31). Next, theinvariance breaks the H1 boundedness of ψn. But we are free to replace each ψnby its rescaling so that ‖ψn‖L2 = 1, without losing its properties (2.31). Then 1 =‖ψn‖2L2 = 2F (ψn) → 2F (ψ), which clearly implies that the limit ψ 6= 0. By (2.30),we may replace ψ by its constant multiple, so that K(ψ) = 0, J(ψ) = H(ψ) ≤ mand ψ 6= 0. Then ψ is a minimizer and m = H(ψ) > 0.

Finally, the invariance gives us L2J(ψ) = 0 and the Lagrange multiplier η maybe nonzero. The equation (2.34) is written in this case

−∆ψ = (ηdβ − 1)[ψ − f ′(ψ)]. (2.38)

Since 〈−∆ψ|ψ〉L2x> 0 and

〈ψ − f ′(ψ)|ψ〉L2x= K0,2/d(ψ)−

∫(D − 2)f(ψ)dx < 0, (2.39)

we have (ηdβ − 1) < 0. Hence there exists λ > 0 such that ψλ solves the staticNLKG (1.39), and it is also a minimizer. �

2.3. H1 critical case; massless threshold. In the H1 critical case (1.28), westill have the (α, β) independence, but mα,β is equal to the massless energy of themassless ground state. This is a consequence of the invariance of the massless energywith respect to the H1 scaling.

Lemma 2.5 (Ground state in H1 critical case). Assume that f satisfies (1.28),and that (α, β) satisfies (1.16). Then mα,β in (1.17) is positive and independent of

(α, β). Moreover mα,β = J (0)(Q) for some Q ∈ H1(Rd) solving the static massless

NLKG (1.41), with the minimal J (0)(Q) among the solutions in H1(Rd).

Proof. Let Hw and Kw be the massless versions of H and K, respectively. Then

m = mw := inf{Hw(ϕ) | ϕ ∈ H1, Kw(ϕ) < 0.}. (2.40)

Indeed, comparing the above with (2.28), we easily get m ≥ mw from Hw ≤ H andKw < K if 2α + dβ > 0. If 2α + dβ = 0, then we may replace K ≤ 0 in (2.28) byK < 0, because for any nonzero ϕ ∈ H1 satisfying K(ϕ) ≤ 0, we have by (2.18)

LK(ϕ) ≤ µK(ϕ)− 2αεµ

d+ 1F (ϕ) < 0, (2.41)

which implies that K(ϕλ) < 0 for all λ > 0, and so the set K < 0 is dense in theminimization set of (2.28). Hence m ≥ mw in this case too.

To prove m ≤ mw, let

ϕν = ϕνd/2−1,−1 (2.42)


denote the H1 invariant scaling. Then K(ϕν) → Kw(ϕ) and H(ϕν) → Hw(ϕ) asν → ∞. Hence if Kw(ϕ) < 0 then K(ϕν) < 0 for large ν, and so m ≤ mw.

Due to the H1 scale invariance, Kwα,β for all (α, β) are constant multiples of the

same functional, and Hw is independent of (α, β), so is the minimization for mw. Infact we have

mw = inf{‖∇ϕ‖2L2/d | ϕ ∈ H1, ‖∇ϕ‖2L2 < ‖ϕ‖2⋆L2⋆}. (2.43)

By the homogeneity and the scaling ϕ 7→ νϕ, it is equal to

inf06=ϕ∈H1

1

d‖∇ϕ‖2L2

[‖∇ϕ‖2L2

‖ϕ‖2⋆L2⋆

] d−22

= inf06=ϕ∈H1

1

d

[‖∇ϕ‖L2

‖ϕ‖L2⋆

]d=

(C⋆S)

−d

d, (2.44)

where C⋆S denotes the best constant for the Sobolev inequality

‖ϕ‖L2⋆ ≤ C⋆S‖∇ϕ‖L2 , (2.45)

which is well known to be attained by the following explicit Q ∈ H1

Q(x) =

[1 +

|x|2d(d− 2)

]− d−22

, (2.46)

which solves (1.41). �

2.4. Exponential case; mass-modified threshold. In the 2D exponential case(1.29), the conclusion is somewhat intermediate between the above two cases. IfC⋆

TM(F ) ≥ 1 then mα,β is achieved by a ground state, but if C⋆

TM(F ) < 1 then we

can still see mα,β as the energy of a ground state to an equation (1.43) where themass is changed to c = min(1, C⋆

TM(F )) ∈ (0, 1).

Lemma 2.6 (Ground state in the exponential case). Assume that f satisfies (1.36)and (1.29), and that (α, β) satisfies (1.16). Then mα,β in (1.17) is independent of

(α, β) and 0 < mα,β ≤ 2π/κ0, where the second inequality is strict if and only if

C⋆TM(F ) > 1. Moreover mα,β = J (c)(Q) with c = min(1, C⋆

TM(F )) for some Q ∈H1(R2) solving the modified static NLKG (1.43) with the minimal J (c)(Q) among

the solutions in H1(R2).

For the proof, we prepare some notations and lemmas. For any functional G onH1(R2) and any A > 0, we introduce the Trudinger-Moser ratio

CATM

(G) := sup{2G(ϕ)‖ϕ‖−2L2 | 0 6= ϕ ∈ H1(R2), ‖∇ϕ‖L2 ≤ A}, (2.47)

the Trudinger-Moser threshold on the H1 norm:

M(G) := sup{A > 0 | CATM(G) <∞}, (2.48)

and the ratio on the threshold:

C⋆TM

(G) := CM(G)TM (G). (2.49)

The growth condition (1.29) together with (1.21) implies

M(Lα,βF ) = M(F ) =√4π/κ0 (2.50)

for any (α, β) satisfying (1.16), by the Trudinger-Moser inequality (2.13). Hence theabove definition of C⋆

TM is consistent with (1.30).


For any functionalG of the formG(ϕ) =∫g(ϕ)dx, and for any sequence (ϕn)n∈N ∈

H1(R2)N, we define its concentration (at x = 0) conc.G((ϕn)n∈N) by

conc.G((ϕn)n∈N) := limε→+0

limn→∞

∫

|x|<ε

g(ϕn)dx. (2.51)

We will use the following compactness by dominated convergence.

Lemma 2.7. Let g, h : R → R be continuous functions satisfying

limu→±∞

|g(u)|h(u)

= 0, limu→0

|g(u)||u|2 = 0. (2.52)

Let ϕn be a sequence of radial functions, weakly convergent to ϕ in H1(R2) such that

{h(ϕn)}n is bounded in L1(R2). Then g(ϕn) → g(ϕ) strongly in L1(R2).

Proof. By assumption (2.52), for any ε > 0 there exist δ > 0 such that

|u| > 1/(2δ) or |u| < 2δ =⇒ |g(u)| < ε(h(u) + |u|2). (2.53)

Then we have∫

|ϕn|>1/(2δ) or |ϕn|<2δ

|g(ϕn)|dx. ε

∫h(ϕn) + |ϕn|2dx. ε. (2.54)

The radial Sobolev inequality ‖r1/2ϕn‖L∞ . ‖ϕn‖1/2L2 ‖∇ϕn‖1/2L2 implies that ϕn(x) areuniformly small for large x. Then the weak convergence together with

ϕn(R1)− ϕn(R2) =

∫ R2

R1

∂rϕn(r)dr (2.55)

implies that ϕn(x) → ϕ(x) for x 6= 0. Then Fatou’s lemma implies∫

|ϕ|>1/(2δ) or |ϕ|<2δ

|g(ϕ)|dx. ε, (2.56)

and the dominated convergence theorem implies

‖g(δ)(ϕn)− g(δ)(ϕ)‖L1 → 0, (n→ ∞) (2.57)

where g(δ) is defined by g(δ)(u) = (1−χδ(u))χ1/δ(u)g(u) using the cut-off defined in(1.23). Combining (2.54), (2.56) and (2.57), we deduce the desired convergence. �

Proof of Lemma 2.6. We start with the exceptional case (d, α) = (2, 0). First, letA > 0 and assume CA

TM(F ) > 1. Then there exists 0 6= ϕ ∈ H1 such that ‖∇ϕ‖L2 ≤A and F (ϕ) > ‖ϕ‖2L2/2. For small ε > 0 we have K0,1((1 − ε)ϕ) < 0, and hencem0,1 ≤ ‖∇(1− ε)ϕn‖2L2/2 < A2/2. Hence m0,1 ≤ M(F )2/2.

Consider the case C⋆TM

(F ) > 1. Then by choosing A = M(F ) in the aboveargument, we get m0,1 < M(F )2/2. Now we take a minimizing sequence for m0,1.By the Schwartz symmetrization and rescalings as in the proof of Lemma 2.4 for(d, α) = (2, 0), we get a sequence of radial functions ψn ∈ H1 such that

‖ψn‖L2 = 1, H0,1(ψn) → m0,1, K0,1(ψn) = 1− 2F (ψn) = 0, (2.58)

and ψn → ψ in H1. Because of m0,1 < M(F )2/2, we can choose some κ ∈(κ0, 2π/m0,1), so that eκ|ψn|2 −1 is bounded in L1 by the Trudinger-Moser inequality

(2.13). Then we can use Lemma 2.7 with ϕn := ψn, g := f and h(u) := eκ|u|2 − 1,


which implies F (ψn) → F (ψ). Hence ψ attains m0,1. After appropriate rescalings,we obtain a ground state Q, as in the proof of Lemma 2.4.

Next consider the case C⋆TM

(F ) ≤ 1. Then for any ψ ∈ H1 satisfying ‖∇ψ‖L2 ≤M(F ) we have K0,1(ψ) ≥ 0. Hence

m0,1 = inf{‖∇ϕ‖2L2/2 | K0,1(ϕ) < 0} ≥ M(F )2/2, (2.59)

and so m0,1 = M(F )2/2. Now we show that there exists ϕ ∈ H1 satisfying

‖∇ϕ‖L2 = M(F ), F (ϕ) = C⋆TM(F )/2, ‖ϕ‖L2 = 1. (2.60)

After rescaling this ϕ, we obtain a ground state Q. However, due to the criticality,we have to approximate the problem by a subcritical one, namely we first prove theexistence of ϕn ∈ H1 satisfying

‖∇ϕn‖L2 ≤ M(F )− 1

n, F (ϕn) = cn/2, ‖ϕn‖L2 = 1 (2.61)

where cn := CM(F )− 1

nTM (F ), then 0 < cn ր C⋆

TM(F ) ≤ 1. Fix n ≫ 1 and let ϕk ∈

H1(R2) be a maximizing sequence for cn (see (2.47)):

‖∇ϕk‖L2 ≤ M(F )− 1

n, F (ϕk) ր cn/2, ‖ϕk‖L2 = 1, (2.62)

where the L2 norm is normalized by the rescaling ϕλ0,1. The Schwartz symmetrization

enables us to assume that ϕk are radial functions, and convergent to some ϕn weaklyin H1, by extracting a subsequence. Moreover, we have F (ϕk) → F (ϕn) = cn/2, by

Lemma 2.7 with g := f and h = eκ|u|2 − 1 for some κ ∈ (κ0, 4π/(M(F )− 1/n)2).

Thus ϕn is a maximizer, which implies that ‖ϕn‖L2 = 1 and

−η∆ϕn = f ′(ϕn)− cnϕn, (2.63)

for a Lagrange multiplier η(n) ∈ R. Multiplying it with ϕn, we obtain

η‖∇ϕn‖2L2 =

∫Df(ϕn)dx− cn‖ϕn‖2L2 =

∫(D − 2)f(ϕn)dx > 0, (2.64)

since (D − 2)f > 0. Hence η > 0, and so Qn(x) := ϕn(η1/2x) ∈ H1 satisfies

‖∇Qn‖L2 ≤ M(F )− 1

n, −∆Qn + cnQn = f ′(Qn). (2.65)

Now consider the limit n→ ∞. The equation forQn implies that 0 = K(cn)0,1 (Qn) =

K(cn)1,−1(Qn), that is

cn‖Qn‖2L2 = 2F (Qn), ‖∇Qn‖2L2 = 2

∫(D − 2)f(Qn)dx ≥ 4F (Qn), (2.66)

where the last inequality follows from (D−4)f ≥ 0. Since ‖∇Qn‖L2 is bounded andcn is positive non-decreasing, we deduce that ‖Qn‖L2 and

∫Df(Qn)dx are bounded

as n → ∞. Hence we may extract a subsequence so that Qn converges to some Qweakly in H1, and then apply Lemma 2.7 with ϕn := Qn, g = f ′ and h := Df .Then f ′(Qn) → f ′(Q) strongly in L1, and so Q solves

−∆Q + cQ = f ′(Q), c := C⋆TM

(F ). (2.67)


This implies that

K(c)0,1(Q) = 〈J (c)′(Q)|L0,1Q〉 = 0, (2.68)

namely 2F (Q) = c‖Q‖2L2. Hence Q is a maximizer for CM(F )TM (F ) with a non-zero

Lagrange multiplier, which implies that ‖∇Q‖L2 = M(F ). Thus J (c)(Q) = M(F )2/2is unique for any solution Q of (2.67).

Next we consider mα,β with α > 0. If m0,1 <M(F )2/2, then there exists a groundstate Q, which satisfies Kα,β(Q) = 0 for all (α, β). Hence mα,β ≤ J(Q) = m0,1.

Otherwise, m0,1 = M(F )2/2 = M(LF )2/2. For any A > M(LF ), there exists asequence ϕn ∈ H1 satisfying

‖∇ϕn‖L2 ≤ A, ‖ϕn‖L2 → 0, LF (ϕn) → ∞. (2.69)

Since K(ϕ) = α‖∇ϕ‖2L2 + (α + β)‖ϕ‖2L2 − LF (ϕ) and α > 0, we can replace eachϕn with ϕn(x/νn) with some νn → +0, so that we have after the rescaling

‖∇ϕn‖L2 ≤ A, K(ϕn) = 0, ‖ϕn‖L2 → 0. (2.70)

Hence mα,β ≤ limn→∞ J(ϕn) ≤ A2/2, and so mα,β ≤ M(LF )2/2 = m0,1. Thus inboth cases we have mα,β ≤ m0,1 ≤ M(F )2/2.

Now suppose that mα,β < m0,1 ≤ M(F )2/2. As in the proof of Lemma 2.4 for(d, α) 6= (2, 0), we may find a sequence of radial ϕn ∈ H1 such that

K(ϕn) = 0, H(ϕn) ց m, (2.71)

and ϕn → ∃ϕ weakly in H1, and pointwise for x 6= 0.Let ψn = ϕn − ϕ. Then ψn → 0 weakly in H1, and so

limn→∞

KQ(ϕn) = limn→∞

KQ(ψn) +KQ(ϕ)

= limn→∞

LF (ϕn) = conc.LF ((ϕn)n) + LF (ϕ), (2.72)

where the second identity is because K(ϕn) = 0, and the last one follows fromϕn(x) → ϕ(x) for x 6= 0 and the radial Sobolev inequality ‖r1/2ϕn‖L∞ . ‖ϕn‖H1.Since H(ϕ) ≤ m by Fatou’s lemma, we have K(ϕ) ≥ 0, otherwise there would besome λ < 0 such that K(ϕλ) = 0 and H(ϕλ) < H(ϕ) ≤ m, a contradiction. ThusKQ(ϕ) ≥ LF (ϕ), and so from (2.72), we deduce

limn→∞

KQ(ψn) ≤ conc.LF ((ϕn)n). (2.73)

Since LF (ϕn) is bounded by (2.72), Lemma 2.7 with hn := (αD + βd)f impliesthat conc.F ((ϕn)n) = 0. Hence by (2.73) and (L− µ)F ≥ 0, we get

limn→∞

KQ(ψn) ≤ conc.(L − µ)F ((ϕn)n) ≤ limn→∞

(L − µ)F (ϕn). (2.74)

On the other hand we have

m = limn→∞

H(ϕn) = limn→∞

HQ(ψn) +HQ(ϕ) + limn→∞

(L − µ)F (ϕn)/µ, (2.75)

where HQ(ψ) := (1 − L/µ)‖ψ‖2H1/2 denotes the quadratic part of H . Combiningthe above two, and discarding HQ(ϕ) ≥ 0, we obtain

limn→∞

‖ψn‖2H1/2 ≤ m <M(F )2/2 = 2π/κ0. (2.76)


Hence applying Lemma 2.7 to ϕn with h(u) := eκ|u|2 − 1 for some κ ∈ (κ0, 2π/m),

we get LF (ϕn) → LF (ϕ), and so ϕ is a minimizer for mα,β. Indeed, we have

eκ|ϕn|2 − 1 ≤ eCκ,κ′ |ϕ|2 − 1 + eκ

′|ψn|2 − 1 (2.77)

for some κ′ ∈ (κ, 2π/m) and constant Cκ,κ′ > 0. Hence h(ϕn) is uniformly bounded

in L1. Recall that for a fixed ϕ ∈ H1, eCκ,κ′ |ϕ|2 − 1 ∈ L1.

Then as in the proof of Lemma 2.4, we obtain a ground state Q with J(Q) =mα,β < m0,1, which is a contradiction since K0,1(Q) = 0. Hence mα,β = m0,1 for all(α, β) in the range (1.16). �

Remark 2.8. In the above argument for (α, β) = (0, 1) in the case C⋆TM

(F ) ≤ 1,we used a priori bounds on the ground state to get the compactness. For generalsequences, we can have concentrating loss of compactness on the kinetic threshold‖∇ϕ‖L2 = M(F ) if and only if f satisfies

lim|u|→∞

e−κ0|u|2|u|2f(u) ∈ (0,∞). (2.78)

The above result implies that the concentration requires more energy than the (mass-modified) ground state. Similar phenomena have been observed in slightly different

settings (either on a bounded domain or on the H1(R2) threshold, where eκ0|u|2

appears as the critical growth instead of eκ0|u|2/|u|2, see [11, 16, 40]). More details

about this issue, including the above concentration compactness, will be addressedin a forthcoming paper [24].

2.5. Parameter independence of the splitting. The (α, β)-independence ofK±α,β follows from that of mα,β and contractivity of K+

α,β.

Lemma 2.9 (Parameter independence of K±). Assume that f satisfies (1.36), andthat (α, β) satisfies (1.16). Then K±

α,β in (1.18) are independent of (α, β).

Proof. Since mα,β is independent of (α, β), we only need to see that the sign ofK is independent under the threshold m. Moreover, we may restrict to the firstcomponent. For any δ ≥ 0, we define K±δ

α,β ⊂ H1 by

K+δα,β = {ϕ ∈ H1 | J(ϕ) < m− δ, Kα,β(ϕ) ≥ 0},

K−δα,β = {ϕ ∈ H1 | J(ϕ) < m− δ, Kα,β(ϕ) < 0}.

(2.79)

Then (u0, u1) ∈ K±α,β if and only if u0 ∈ K±δ

α,β with δ = ‖u1‖2L2/2. In addition, the

disjoint union K+δα,β ∪ K−δ

α,β is already independent of α and β. Hence it suffices to

show the independence of K+δα,β.

First we consider the interior exponents satisfying 2α+dβ > 0 and 2α+(d−2)β >0. Then K+δ

α,β is contracted to {0} by the rescaling ϕ 7→ ϕλ with 0 ≥ λ→ −∞. Thisis due to the following facts

(1) K(ϕλ) > 0 is preserved as long as J(ϕλ) < m, by the definition of m.(2) J(ϕλ) does not increase as λ decreases, as long as LJ(ϕλ) = K(ϕλ) > 0.(3) ϕλ → 0 in H1 as λ→ −∞, since 2α+ dβ > 0 and 2α+ (d− 2)β > 0.

In particular, J cannot be negative on K+α,β, and so K+δ

α,β = ∅ for δ ≥ m. For

0 ≤ δ < m, both K±δα,β are open in H1. It follows for K−δ from the definition, and


for K+δ from the facts that J(ϕ) < m and K(ϕ) = 0 imply ϕ = 0, and that aneighborhood of 0 is contained in K+δ, which follows from (2.7), (2.8) or (2.14).Then the above argument of the scaling contraction shows that K+δ

α,β is connected.

Hence each K+δα,β cannot be separated by K+δ

α′,β′ and K−δα′,β′ with any other (α′, β ′) in

the interior range. Since K+δα,β ∩ K+δ

α′,β′ ∋ 0, we conclude that K+δα,β = K+δ

α′,β′.

Finally for (α, β) on the boundary 2α + dβ = 0 or 2α + (d − 2)β = 0, take asequence (αn, βn) in the interior converging to (α, β). Then Kαn,βn → Kα,β, and so

K±δα,β ⊂

⋃

n

K±δαn,βn

. (2.80)

Since the right hand side is independent of the parameter, so is the left. �

2.6. Variational estimates. We conclude this section with a few estimates on theenergy-type functionals, which will be important in the proof of the blow-up andthe scattering. We start with the easy observation that the free energy and thenonlinear energy are equivalent in the set K+.

Lemma 2.10 (Free energy equivalence in K+). Assume that f satisfies (1.36). Thenfor any (u0, u1) ∈ H1(Rd)× L2(Rd) we have

K1,0(u0) ≥ 0 =⇒{J(u0) ≤ ‖u0‖2H1

x/2 ≤ (1 + d/2)J(u0),

E(u0, u1) ≤ EQ(u0, u1) ≤ (1 + d/2)E(u0, u1).(2.81)

Proof. Since (D − 2 − c)f(u) ≥ 0 with c := 4/d > 0 by (1.21), we have for any(u0, u1) ∈ H1 × L2,

K1,0(u0) = ‖u0‖2H1x− (2 + c)F (u0)−

∫(D − 2− c)f(u0)dx

≤ (2 + c)J(u0)− c‖u0‖2H1x/2 = (2 + c)E(u0, u1)− cEQ(u0, u1)− ‖u‖2L2

x,

(2.82)

and hence we obtain the desired estimate. �

In the 2D exponential case, we have a sharper bound on the derivatives, whichimplies that K+ is in the subcritical regime for the Trudinger-Moser inequality.

Lemma 2.11 (Subcritical bound in K+ in the 2D exponential case). Assume that

f satisfies (1.36) and (1.29). Then for any (u0, u1) ∈ K+ we have

‖∇u0‖2L2 + ‖u1‖2L2 < 2m ≤ M(F )2 = 4π/κ0. (2.83)

Proof. Since K0,1(u0) ≥ 0, we have

‖∇u0‖2L2 + ‖u1‖2L2 ≤ ‖∇u0‖2L2 + ‖u1‖2L2 +K0,1(u0) = 2E(u0, u1) < 2m. (2.84)

�

The next estimate gives a lower bound on |K| under the threshold m, which willbe important both for the blow-up and for the scattering.

Lemma 2.12 (Uniform bounds on K). Assume that f satisfies (1.21), and that

(α, β) satisfies (1.16) and (d, α) 6= (2, 0). Then there exists δ > 0 determined by

(α, β), d and ε in (1.21), such that for any ϕ ∈ H1 with J(ϕ) < m we have

Kα,β(ϕ) ≥ min(µ(m− J(ϕ)), δKQα,β(ϕ)) or Kα,β(ϕ) ≤ −µ(m− J(ϕ)). (2.85)


Note that if (d, α) = (2, 0) then the conclusion is false, since in that caseK(ϕλα,β) =

edβλK(ϕ) → 0 as λ → −∞, while J(ϕλ) is away from m, since it is decreasing ifK(ϕ) > 0 and J(ϕλ) ր H(ϕ) < m if K(ϕ) < 0.

Proof. Wemay assume ϕ 6= 0. Let j(λ) = J(ϕλ) and n(λ) = F (ϕλ), where ϕλα,β = ϕλ

is the rescaling (1.13). Then j(0) = J(ϕ) and j′(0) = K(ϕ), and (2.18) implies

j′′ ≤ (µ+ µ)j′ − µµj − 2αε

d+ 1n′. (2.86)

First we consider the case K(ϕ) < 0. By Lemma 2.1 together with (2.3), thereexists λ0 < 0 such that j′(λ) < 0 for λ0 < λ ≤ 0 and j′(λ0) = 0. For λ0 ≤ λ ≤ 0 wehave from (2.16),

(µ+ µ)j′ − µµj ≤ µj′. (2.87)

Inserting this in (2.86) and integrating it, we get∫ 0

λ0

j′′(λ)dλ ≤ µ

∫ 0

λ0

j′(λ)dλ, (2.88)

and hence

K(ϕ) = j′(0) ≤ µ(j(0)− j(λ0)). (2.89)

Since K(ϕλ0) = 0 and ϕλ0 6= 0, we have j(λ0) = J(ϕλ0) ≥ m. Thus we obtain

K(ϕ) ≤ −µ(m− J(ϕ)). (2.90)

Next we consider the case K(ϕ) > 0. If

(2µ+ µ)K(ϕ) ≥ µµJ(ϕ) +2αε

d+ 1LF (ϕ), (2.91)

then applying (2.81) to the first term on the right hand side, and K = KQ−LF tothe second one, we get

[2µ+ µ+

2αε

d+ 1

]K(ϕ) ≥

µµ

2 + d‖ϕ‖2H1 +

2αε

d+ 1KQ(ϕ), (2.92)

and so K(ϕ) ≥ δKQ(ϕ) for some δ > 0, since µ > 0 or α > 0. If (2.91) fails, then

(2µ+ µ)j′ < µµj +2αε

d+ 1n′, (2.93)

at λ = 0, and so from (2.86),

j′′ < −µj′. (2.94)

Now let λ increase. As long as (2.93) holds and j′ > 0, we have j′′ < 0 and so j′

decreases and j increases. Also by (2.18) and (2.16) we have

n′′ ≥ (µ+ µ)n′ − µµn ≥ µn′ ≥ µ2n > 0. (2.95)

Hence (2.93) is preserved until j′ reaches 0. It does reach at finite λ0 > 0, becausethe right hand side of (2.86) is negative and decreasing as long as j′ > 0. Nowintegrating (2.94) we obtain

K(ϕ) = j′(0) ≥ µ(j(λ0)− j(0)) ≥ µ(m− J(ϕ)), (2.96)


where we used that J(ϕλ0) ≥ m which follows from K(ϕλ0) = 0 and ϕλ0 6= 0. �

3. Blow-up

Here we prove the blow-up part of Theorem 1.1. The idea is essentially due toPayne-Sattinger [39], but we give a full proof for convenience. We will use that K−

is stable under the flow.By contradiction we assume that the solution u exists for all t > 0. The proof for

t < 0 is the same and omitted. Let

y(t) := ‖u(t, x)‖2L2x(R

d). (3.1)

Multiplying the equation with u, and using (2.82), we get

y = 2‖u‖2L2 − 2K1,0(u) ≥ (4 + c)‖u‖2L2 − 2(2 + c)E(u) + c‖u‖2H1, (3.2)

for some c > 0. Sine u(t) ∈ K−, Lemma 2.12 implies that there is some positiveδ ≤ −K1,0(u(t)). Thus for all t > 0 we have

y(t) ≥ 2δ > 0, (3.3)

and so y(t) = ‖u(t)‖2L2 → ∞ as t→ ∞. Going back to (3.2), and using Schwarz, wededuce that for large t

y ≥ (4 + c)‖u‖2L2 >4 + c

4

y2

y, (3.4)

therefore

(y−c/4)tt = − c4y−c/4−2

[yy − 4 + c

4y2]< 0, (3.5)

which contradicts that y → ∞.

4. Global space-time norm

In this section we introduce Strichartz-type estimates and a perturbation lemmafor global space-time bounds of the solution.

The inhomogeneity of the Klein-Gordon equation makes the exponents a bit morecomplicated than the case of wave or Schrodinger equation. In the H1 critical case,we get another complication in higher dimensions, due to the fact that we have toestimate the difference of solutions in some Sobolev (or Besov) spaces with positiveregularity but the nonlinearity is not twice differentiable2. This is not a problemin the subcritical case, where we are allowed to lose small regularity, so that wecan estimate the difference in some Lp spaces and then interpolate. This technicalissue was solved in the pure critical case in [36] by using space-time norms withexponents away from the admissible region for the standard Strichartz estimate,which was later called “exotic Strichartz estimates” in the Schrodinger case [46].

Here we have a further complication by the presence of lower powers, for whichwe need the exotic Strichartz for the Klein-Gordon equation. Note that it is not abig trouble in the Schrodinger case (see [47]), because the same Strichartz estimateis used both for higher and lower powers. In the Klein-Gordon case, in contrast, we

2The problem is not on the local regularity of the nonlinearity (at u = 0), but rather on theglobal Holder continuity for fL.


have to use different Strichartz norms, with better regularity for higher powers andwith better decay for lower powers. It is easy in the standard Strichartz estimate,where we can freely mix different norms by the duality argument, but this does notwork for the exotic Strichartz estimate, which uses exponents away from the duality.Hence we are forced to use a common exponent for different powers, which makesour estimates much more involved. In particular, when we have both the H1 criticaland the L2 critical powers, we need three steps to close our estimates.

4.1. Reduction to the first order equation. To simplify the notation, we rewriteNLKG in the first order equation. With any real-valued function u(t, x), we associatethe complex-valued function ~u(t, x) by

~u = 〈∇〉u− iu, u = 〈∇〉−1ℜ~u. (4.1)

This relation u↔ ~u will be assumed for any space-time function u throughout thispaper. Here we use i purely for notational convenience, and we could use a vectorform instead3, especially if u is originally complex-valued. The free and nonlinearKlein-Gordon equations are given by

(�+ 1)u = 0 ⇐⇒ (i∂t + 〈∇〉)~u = 0,

(�+ 1)u = f ′(u) ⇐⇒ (i∂t + 〈∇〉)~u = f ′(〈∇〉−1ℜ~u), (4.2)

and the free energy is given by EQ(u) = ‖~u‖2L2x/2. We denote

E(ϕ) := ‖ϕ‖2L2x/2− F (〈∇〉−1ℜϕ),

Kα,β(ϕ) := KQα,β(〈∇〉−1ϕ) +KN

α,β(〈∇〉−1ℜϕ).(4.3)

Remark that

E(~u(t)) = E(u; t), K(~u(t)) ≥ K(u(t)), (4.4)

where the equality in the latter holds if and only if u(t) = 0. Nevertheless, theinvariant set K+ = K+

α,β for ~u is given by

K+ := {ϕ ∈ L2(Rd) | E(ϕ) < m, K(ℜ〈∇〉−1ϕ) ≥ 0}= {ϕ ∈ L2(Rd) | E(ϕ) < m, K(ϕ) ≥ 0}.

(4.5)

The second identity (the first one is definition) is proved as follows. Let ϕ ∈ L2(Rd)

satisfy E(ϕ) < m and K(ℜ〈∇〉−1ϕ) < 0. Let ψ1 = ℜ〈∇〉−1ϕ and ψ2 = ℑ〈∇〉−1ϕ.Then Lemma 2.12 implies that

K(ψ1) ≤ −µ(m− J(ψ1)) < −µ‖ψ2‖2H1x/2 ≤ −KQ(ψ2), (4.6)

so K(ϕ) = K(ψ1) +KQ(ψ2) < 0. Hence under the condition E(ϕ) < m, the signs

of K(ψ1) and K(ϕ) are the same, which proves (4.5).

3We chose the complex form rather than the vector one, in order to avoid adding a subscript,for this notation will be applied mostly to sequences.


4.2. Strichartz-type estimates and exponents. Here we recall the Strichartzestimate for the free Klein-Gordon equation, introducing some notation for thespace-time norms and special exponents.

With any triplet (b, c, σ) ∈ [0, 1]2×R and any q ∈ (0,∞], we associate the followingBanach function spaces on I × Rd for any interval I:

[(b, c, σ)]q(I) := L1/bt (I;Bσ

1/c,q(Rd)), [(b, c, σ)]0(I) := L

1/bt (I;L1/c(Rd)),

[(b, c, σ)]•q(I) := L1/bt (I; Bσ

1/c,q(Rd)),

(4.7)

where Bsp,q and B

sp,q respectively denote the inhomogeneous and homogeneous Besov

spaces, and the following characteristic numbers with a parameter θ ∈ [0, 1]:

regθ(b, c, σ) := σ − (1− 2θ/d)b− d(c− 1/2),

strθ(b, c, σ) := 2b+ (d− 1 + θ)(c− 1/2),

decθ(b, c, σ) := b+ (d− 1 + θ)(c− 1/2).

(4.8)

θ = 0, 1 correspond respectively to the wave and the Klein-Gordon equations. regθ

indicates the regularity of the space, while strθ and decθ indicate the space-timedecay, corresponding respectively to the Strichartz and the Lp−Lq decay estimates.We denote the regularity change and the duality in Hs−1/2 (here −1/2 takes accountof one regularity gain in the wave equation) respectively by

(b, c, σ)s := (b, c, s), (b, c, σ)∗(s) := (1− b, 1− c,−σ + 2s− 1). (4.9)

Given a real number s, we say Z = (Z1, Z2, Z3) is Strichartz s-admissible if forsome θ ∈ [0, 1] we have

0 ≤ Z1 ≤ 1/2, 0 ≤ Z2 < 1/2, regθ(Z) ≤ s, strθ(Z) ≤ 0. (4.10)

We avoid the endpoint Z2 = 1/2 to mix different θ. The Strichartz estimates read

Lemma 4.1 (see [9, 17, 31]). For any s ∈ R, let Z and T be s-admissible. Then

for any space-time function u(t, x), any interval I ⊂ R, and any t0 ∈ I, we have

‖u‖[Z]2(I). ‖u(t0)‖Hs + ‖u(t0)‖Hs−1 + ‖u−∆u+ u‖[T ∗(s)]2(I), (4.11)

where the implicit constant does not depend on I or t0.

The “exotic Strichartz estimate” is given for the Klein-Gordon equation by

Lemma 4.2. Let Z, T ∈ R3 satisfy for some θ ∈ [0, 1]

regθ(Z) ≤ regθ(T ) + 2, strθ(Z) ≤ strθ(T )− 2, 0 < Z1, T1 < 1,

decθ(Z) < 0 < decθ(T )− 1, 0 <1

2− Z2, T2 −

1

2<

1

d− 1 + θ.

(4.12)

Then we have for any interval I ⊂ R, t0 ∈ I, and u(t, x) satisfying u(t0) = u(t0) = 0,

‖u‖[Z]2(I). ‖u−∆u+ u‖[T ]2(I). (4.13)

Proof. The wave case θ = 0 was essentially proved in [36, Lemma 7.4], where theborderline case str0(Z) = str0(T ) − 2 was excluded for the real interpolation toimprove the Besov exponent 2. Here we discard that improvement, restoring theborderline case, which is needed for the lower critical power p1 = 4/d.


The proof is rather immediate from the standard Strichartz estimate and the Lp

decay estimate. Indeed, if strθ(Z) = 0 = strθ(T ) − 2 and regθ(Z) = regθ(T ) + 2,then the above estimate is nothing but Strichartz. If moreover Z2 + T2 = 1, thenthe estimate directly follows from the Lp decay and Hardy-Littlewood-Sobolev

‖∫ t

t0

〈∇〉−1e±i(t−s)〈∇〉h(s)ds‖[Z]2(I)

. ‖∫ t

t0

|t− s|−2Z1‖h(s)‖B

T31/T2,2

ds‖L1/T1 (I). ‖h‖[T ]2(I).(4.14)

This estimate can be translated in the time and the regularity exponents as

Z 7→ Z ′ = Z + (b, 0, s), T 7→ T ′ = T + (b, 0, s) (4.15)

for any s ∈ R and b ∈ (−1/2, 1/2), as long as 0 < Z ′1, T

′1 < 1. By the complex

interpolation for those estimates and the standard Strichartz estimate, we obtainthe desired estimate in the case strθ(Z) = strθ(T )− 2 and regθ(Z) = regθ(T )+ 2. Itis extended to the remaining cases (with inequality in these relations) by the Sobolevembedding. �

The following interpolation is convenient to switch from some exponents to others,

Lemma 4.3. Let Z,A,B, C ∈ [0, 1]× R and θ ∈ [0, 1]. Assume that A1 < Z1 < B1

and one of the followings

(1) min(strθ(A), strθ(B), strθ(C)) ≥ strθ(Z) and min(regθ(A), regθ(B)) > regθ(Z)(2) min(strθ(A), strθ(B)) > strθ(Z) andmin(regθ(A), regθ(B), regθ(C)) ≥ regθ(Z).

Then there exist α, β, γ ∈ (0, 1) satisfying α + β + γ = 1 and for all q ∈ (0,∞] wehave the interpolation inequality

‖u‖[Z]q . ‖u‖α[A]∞‖u‖β[B]∞

‖u‖γ[C]∞. (4.16)

Proof. Since A1 < Z1 < B1, for any 0 < θ2 ≪ 1 there exists θ1 ∈ (0, 1) such that

(1− θ2)((1− θ1)A1 + θ1B1) + θ2C1 = Z1. (4.17)

Let Z := (1− θ2)((1− θ1)A+ θ1B) + θ2C. Then from the assumption we have

strθ(Z) ≥ strθ(Z), regθ(Z) ≥ regθ(Z), (4.18)

which imply Z2 ≥ Z2 and Z3 − dZ2 ≥ Z3 − dZ2, and so we have the Sobolev

embedding [Z]q ⊂ [Z]q. In the first case, we have regθ(Z) > regθ(Z) and so

[[[A]∞, [B]∞]θ1 , [C]∞]θ2 = [Z]∞ ⊂ [Z]q. (4.19)

The desired inequality follows from that for the complex interpolation.It remains to prove in the second case. By the real interpolation in the Besov

space in x and Holder in t, we have for all 0 < δ ≪ 1,

‖u‖[Z]q . ‖u‖1/2[Z+]∞‖u‖1/2[Z−]∞

, Z± := Z ± δ(1, 0, 1− 2θ/d). (4.20)

Let 0 < ε≪ 1 satisfy ε(B1 −A1)(1− θ2) = δ and

Z± := (1− θ2)((1− θ1 ∓ ε)A+ (θ1 ± ε)B) + θ2C. (4.21)


Then from the assumption and the definition of Z± and ε, we have

strθ(Z±) > strθ(Z±), regθ(Z±) ≥ regθ(Z±) = regθ(Z), (4.22)

when ε > 0 is small. Hence we have the Sobolev embedding

[[[A]∞, [B]∞]θ1±ε, [C]∞]θ2 = [Z±]∞ ⊂ [Z±]∞, (4.23)

where the left hand side is a nested complex interpolation space. Now the conclusionfollows from the interpolation inequality. �

4.3. Global perturbation of Strichartz norms. Now we fix a few particularexponents. Define H,W,K by

H :=

(0,

1

2, 1

), W :=

(d− 1

2(d+ 1),W1,

1

2

), K :=

(d

2(d+ 2), K1,

1

2

). (4.24)

Then [H ]2 = L∞t H

1x is the energy space, while W and K are 1-admissible, diagonal

and boundary exponents respectively for the wave (θ = 0) and the Klein-Gordon(θ = 1) equations:

1 = reg0(H) = reg1(H) = reg0(W ) = reg1(K),

0 = str0(H) = str1(H) = str0(W ) = str1(K).(4.25)

Let eq(u) denote the left hand side of NLKG

eq(u) := utt −∆u+ u− f ′(u). (4.26)

Recall the convention u ↔ ~u in Section 4.1 to switch to the first order equations.We will treat the H1 critical case (1.28) together with the subcritical case. SincefS(u) is for small |u| and fL(u) for large |u|, we may freely lower p1 in (1.25) andraise p2 in (1.26). Hence we assume (1.25) with

2⋆ − 2 =4

d< p1 <

4(d+ 1)

(d+ 2)(d− 1), (4.27)

and we assume either d = 1, (1.29) or (1.26) with

4d− 2

d(d− 2)< p2 ≤ 2⋆ − 2. (4.28)

Before the main perturbation lemma, we see that [H ]2 ∩ [W ]2 ∩ [K]2 is enough tobound the full Strichartz norms of the solutions.

Lemma 4.4. Assume that f satisfies (1.36). Let Z, T and U be 1-admissible. In

the 2D exponential case (1.29), let Θ ∈ (0, 1). Then there exist a constant C1 > 0and a continuous function C2 : (0,∞) → (0,∞) such that for any interval I, anyt0 ∈ I and any w(t, x), we have

‖w‖[Z]2(I) ≤ C1‖~w(t0)‖L2x+ C1‖eq(w)‖([T ∗(1)]2+[U∗(1)]2)(I)

+ C2(‖w‖([H]2∩[W ]2∩[K]2)(I)),(4.29)

provided, in the exponential case, that

supt∈I

κ0‖∇w‖2L2x≤ 4πΘ. (4.30)

We remark that (4.30) is needed only in the exponential case.


Proof. We may assume Θ > 1/2 without losing any generality. We introduce thenew exponents M ♯ and X by

M ♯ :=2

p2(d+ 1)(1, 1, 0), X := (ν, 0, ν − ν2), (4.31)

with some ν ∈ (0, 1/10) satisfying Θ < (1− ν)2, where M ♯ is used only if d ≥ 2 andX only in the exponential case. In either case we have

0 > str0(M ♯), str0(X), 1 ≥ reg0(M ♯), 1 > reg0(X), 0 < M ♯1, X1 < W1. (4.32)

Hence by Lemma 4.3(1), we have

‖w‖[M♯]2(I) + ‖w‖[X]2(I) . ‖w‖([H]2∩[W ]2∩[K]2)(I). (4.33)

The Strichartz estimate gives

‖w‖[Z]2(I). ‖~w(t0)‖L2x+ ‖eq(w)‖([T ∗(1)]2+[U∗(1)]2)(I)

+ ‖f ′(w)‖([K∗(1)]2+[W ∗(1)]2+L1tL

2x)(I)

.(4.34)

By the standard nonlinear estimate we have

‖f ′S(w)‖[K∗(1)]2(I) . ‖w‖[K]2(I)‖w‖4/d[K]0(I)

, (4.35)

and in the subcritical/critical cases

‖f ′L(w)‖[W ∗(1)]2(I). ‖w‖[W ]2(I)‖w‖p2[M♯]0(I)

. (4.36)

In the exponential case, there are κ > κ0 and µ > 0 such that

supt∈I

κ‖w‖2H1µ≤ 4πΘ′, (4.37)

where Θ′ := (1 + Θ)/2 < 1 and

‖ϕ‖H1µ:= ‖∇ϕ‖2L2

x+ µ‖ϕ‖2L2

x. (4.38)

Then we have

‖f ′L(w)‖L2

x. ‖|w|(eκ|w|2 − 1)‖L2

x. ‖w‖L∞

x‖eκ|w|2 − 1‖1/2L1

x‖eκ|w|2‖1/2L∞

x, (4.39)

where the second factor is bounded by Trudinger-Moser

‖eκ|w|2 − 1‖L1x. ‖w‖2L2/(1−Θ′), (4.40)

and the third factor is bounded by the following log-interpolation inequality [21,Theorem 1.3]: for any α ∈ (0, 1), λ > 1/(2πα) and µ > 0, there is C > 0 such that

‖ϕ‖2L∞(R2) ≤ λ‖ϕ‖2H1µ(R

2)

[C + log(1 + ‖ϕ‖Cα(R2)/‖ϕ‖H1

µ(R2))], (4.41)

for any ϕ ∈ H1 ∩ Cα(R2), where Cα = Bα∞,∞ denotes the Holder space. Plugging

this with α := ν − ν2 into the exponential, we get

‖eκ|w|2‖L∞x. (1 + ‖w‖Cα

x/‖w‖H1

µ)λκ‖w‖2

H1µ . (1 + κ‖w‖2Cα

x/Θ′)2πλΘ

′

, (4.42)

where λ > 0 is chosen so that

1 < 2πλα, (2πλΘ′ + 1)ν = 1. (4.43)


Since fL vanishes for small |u|, we may assume ‖w‖Cαx& ‖w‖L∞

x& 1. Hence

‖eκ|w|2‖L∞x. ‖w‖4πλΘ′

Cαx

= ‖w‖2(1/ν−1)Cα

x, (4.44)

and plugging this into (4.39), we get

‖f ′L(w)‖L1

tL2x. ‖w‖

L1/νt L∞

x‖w‖L∞

t L2x‖w‖1/ν−1

L1/νt Cα

x

. ‖w‖1/ν[X]2‖w‖[H]2. (4.45)

�

Lemma 4.5. Assume that f satisfies (1.36). Let Z, T , U and V be 1-admissible

and reg0(V ) = 1. In the exponential case (1.29), let Θ ∈ (0, 1). Then there are

continuous functions ε0, C0 : (0,∞)2 → (0,∞) such that the following holds: Let

I ⊂ R be an interval, t0 ∈ I and ~u, ~w ∈ C(I;L2(Rd)). Let ~γ0 = ei〈∇〉(t−t0)(~u− ~w)(t0)and assume that for some A,B > 0 we have

‖~u‖L∞t (I;L2

x) + ‖~w‖L∞t (I;L2

x) ≤ A, (4.46)

‖w‖[W ]2(I)∩[K]2(I) ≤ B, (4.47)

‖(eq(u), eq(w))‖([T ∗(1)]2+[U∗(1)]2)(I) + ‖γ0‖[V ]∞(I) ≤ ε0(A,B), (4.48)

and in the exponential case,

supt∈I

κ0 max(‖∇u‖2L2x, ‖∇w‖2L2

x) ≤ 4πΘ. (4.49)

Then we have

‖u‖[Z]2(I) ≤ C0(A,B). (4.50)

Remark 4.6. (4.49) is needed only in the exponential case. The above lemma remainsvalid in the lower critical case p1 = 4/d = 2⋆ − 2, if we assume in addition that

‖γ0‖[K]0(I) ≤ ε0(A,B). (4.51)

We will indicate the necessary modifications in the proof.

Proof of Lemma 4.5. We restrict p1, p2 as in (4.27) and (4.28), without losing anygenerality. In the following, C(·, . . . ) denotes arbitrary positive constants which maydepend continuously on the indicated parameters. Let δ ∈ (0, 1) be a fixed smallnumber, whose smallness will be specified by the following arguments. Let

e := eq(u)− eq(w), γ := u− w. (4.52)

Then we have the equation for the difference

γ −∆γ + γ = f ′(w + γ)− f ′(w)− e, ~γ(t0) = ~γ0(t0). (4.53)

First note that by Lemma 4.4, we have the full Strichartz norms on w.Next we estimate the difference u − w in the easier case d ≤ 4. We define new

exponents S, L and a space X by

[S]0 := Lp1+1t L2(p1+1)

x , [L]0 := Lp2+1t L2(p2+1)

x ,

X :=

[S]0 (d = 1)

[S]0 ∩ [X ]2 (1.29),

[S]0 ∩ [L]0 (otherwise).

(4.54)


Thanks to the restrictions (4.27) and (4.28), we have

0 > str1(S), str0(L), 1 > reg1(S), reg0(L). (4.55)

Hence by Lemma 4.3(2) with C := V , we get for some θ1, θ2 ∈ (0, 1),

‖γ0‖X (I).A1−θ1εθ10 + A1−θ2εθ20 . (4.56)

If p1 → 4/d, then str0(S) → 0, and we would need the smallness in [K]0(I).Since w ∈ X (I) by Lemma 4.4, there exists a partition of the right half of I:

t0 < t1 < · · · < tn, Ij = (tj, tj+1), I ∩ (t0,∞) = (t0, tn) (4.57)

such that n ≤ C(A,B, δ) and

‖w‖X (Ij) ≤ δ (j = 0, . . . , n− 1). (4.58)

We omit the estimate on I ∩ (−∞, t0) since it is the same by symmetry.Let γj be the free solution defined by

~γj := ei〈∇〉(t−tj )~γ(tj). (4.59)

Then the Strichartz estimate applied to the equations of γ and γj+1 implies

‖γ − γj‖X (Ij) + ‖γj+1 − γj‖X (R). ‖f ′(w + γ)− f ′(w)‖L1tL

2x(Ij)

+ ‖e‖([U∗(1)]2+[T ∗(1)]2)(Ij).(4.60)

The nonlinear difference is estimated as follows. For smaller |u|, we have by Holder

‖f ′S(w + γ)− f ′

S(w)‖L1tL

2x. ‖(w, γ)‖p1[S]0‖γ‖[S]0, (4.61)

and for larger |u| for d ≥ 2 in the subcritical/critical cases,

‖f ′L(w + γ)− f ′

L(w)‖L1tL

2x. ‖(w, γ)‖p2[L]0‖γ‖[L]0. (4.62)

If d = 1, let C(ν) = sup|u|≤ν |f ′′L(u)|/|u|p1. Then we have

‖f ′L(w + γ)− f ′

L(w)‖L1tL

2x.C(‖w‖L∞

t,x+ ‖γ‖L∞

t,x)‖(w, γ)‖p1[S]0‖γ‖[S]0

.C(‖(w, γ)‖L∞t H1

x)‖(w, γ)‖p1[S]0‖γ‖[S]0.

(4.63)

In the exponential case, there exist κ > κ0 and µ > 0 such that (4.37). Let wθ =w + θγ = (1 − θ)w + θu for θ ∈ [0, 1]. Then we have κ‖wθ‖2H1

µ≤ 4πΘ′, where

Θ′ = (1+Θ)/2 and H1µ is defined in (4.38). In the same way as for (4.45), we obtain

‖f ′L(w + γ)− f ′

L(w)‖L1tL

2x≤∫ 1

0

‖f ′′L(wθ)γ‖L1

tL2xdθ

. supθ∈[0,1]

‖wθ‖[H]2‖wθ‖1/ν−1[X]2

‖γ‖[X]2 .A‖(w, γ)‖1/ν−1[X]2

‖γ‖[X]2.(4.64)

Thus in all cases, assuming

‖γ‖X (Ij) ≤ δ ≪ 1, (j = 0, . . . , n− 1), (4.65)

where the smallness depends on A (and Θ), we get

‖γ‖X (Ij) + ‖γj+1‖X (tj+1,tn) ≤ C‖γj‖X (tj ,tn) + ε0, (4.66)


for some absolute constant C ≥ 2. Then by (4.56) and iteration in j we get

‖γ‖X (I). (2C)n(A1−θ1εθ10 + A1−θ2εθ20 ) ≤ C(A,B)(εθ10 + εθ20 ). (4.67)

Choosing ε0(A,B) sufficiently small, we can make the last bound much smaller thanδ, and thus the assumption (4.65) is justified by continuity in t and induction on j.Then repeating the estimate (4.60) once more, we can estimate the full Strichartznorms on γ, which implies also the bound on u.

Next we estimate the difference u − w in the harder case d ≥ 5, where we need

the new exponents M , M , N , N , R, Q, P , and Y defined by

M =2

d+ 1

[1

p2(1− d, 2, 0) +

d− 2

4(d,−1, 0)

],

N =2

d+ 1

[(1

2,d− 1

4, 1

)+

(1− d− 2

4p2

)(−d, 1, 0)

],

M =M +2

p2(d+ 1)(0, 1/d, 1), N = N − 2

d+ 1(0, 1/d, 1),

Q =(1, 2, 2)

p1(d+ 1), P =

(4, d− 1, 4)

2(d+ 1), Y =

(6, d+ 3, 4)

2(d+ 1),

R =

((d+ 4)

2(d+ 2)(p1 + 1), R1,

1

2

).

(4.68)

In the case p2 > 1, we need another exponent

M := M +2(p2 − 1)

p2(d+ 1)(0, 1/d, 1), (4.69)

and if p2 ≤ 1 then we put M = M . Note that p1 < 1 under (4.27) for d ≤ 5. Thenwe have the sharp Sobolev embedding

[M ]q ⊂ [M ]q ⊂ [M ]q, [N ]q ⊂ [N ]q, (4.70)

and nonlinear and interpolation relations

R + p1R0 = K∗(1), R = (1− α)W + αK, M ♯ = (1− β)W 0 + βR0, (4.71)

for some α, β ∈ (0, 1), thanks to (4.27) and (4.28). Y is a non-admissible exponentsatisfying

Y = N + p2M = N + p2M = P + p1Q0 = P 0 + p1Q, (4.72)

where the second and the last identities follow from P3 = p1Q3, N3 = p2M3, and theabove sharp embeddings. If p2 > 1, we have in addition

Y = N + M + (p2 − 1)M. (4.73)


Moreover, these exponents satisfy (when d ≥ 5)

1 = reg0(N) = − reg0(Y ) ≥ reg0(M), 1 > reg1(Q), reg1(P ),− reg1(Y ),

0 > str0(M), str0(N), str1(Q), str1(P ),

str0(N) ≤ str0(Y )− 2, str1(P ) = str1(Y )− 2,

0 ≤ M1, M2, Q1, Q2, R1 < 1/2, 1 < dec0(Y ), dec1(Y ),

Y2 <1

2+

1

d, N2 >

1

2− 1

d− 1, P2 >

1

2− 1

d.

(4.74)

reg0(M) = 1 only if p2 = 2⋆ − 2 = 4/(d− 2). Lemma 4.3(1) implies that

‖w‖([Q]2p1∩[M ]2∩[M ]2p2)(I). ‖w‖([H]2∩[K]2∩[W ]2)(I) .A+B. (4.75)

As before, we divide I ∩ (t0,∞) into t0 < · · · < tn, n ≤ C(A,B) such that

‖w‖([Q]2p1∩[M ]2∩[M ]2p2∩[K]2∩[W ]2)(Ij)≤ δ ≪ 1, (j = 0, . . . , n− 1). (4.76)

We also introduce the following spaces:

Y0 := [W ]0 ∩ [R]0, Y := [N ]2 ∩ [P ]2, Y := [W ]2 ∩ [K]2,

Y∗0 := [W ∗(1)]0 + [K∗(1)]0, Y∗ := [W ∗(1)]2 + [K∗(1)]2.

(4.77)

Our proof for d ≥ 5 consists of three steps:

(1) We estimate γ in Y0, assuming it is bounded in some norm similar to (4.76).Here we can use the standard Strichartz because the estimates do not containspatial derivative.

(2) We estimate γ in Y, under the same assumption on γ. Here we use the exoticStrichartz.

(3) We estimate u in Y by using the bounds in [N ]2 ∩ [R]0. The assumption inthe previous steps is justified once we get a better bound.

Actually we could skip the first step, by using interpolation in the last step to bound[R]0 by the other norms. However, if p1 = 4/d the lower critical power, then R = Kand the first step becomes necessary.

Assuming that

‖γ‖([Q]2p1∩[M ]2p2∩[R]0∩[M♯]0)(Ij)

≤ δ (j = 0, . . . , n− 1), (4.78)

we have by Strichartz and Holder (since W 0 and R0 are 1/2-admissible)

‖γ − γj‖Y0(Ij) + ‖γj+1 − γj‖Y0(R)

. ‖f ′(w + γ)− f ′(w)‖Y∗0 (Ij)

+ ‖e‖Y∗(Ij)

. ‖(w, γ)‖p1[R]0(Ij)‖γ‖[R]0(Ij) + ‖(w, γ)‖p2[M♯]0(Ij)

‖γ‖[W ]0(Ij) + ε0

. δp1‖γ‖Y0(Ij) + ε0,

(4.79)

where we used (4.76) and (4.78). By Lemma 4.3(2), we have

‖γ0‖Y0(I).A1−θ3εθ30 + A1−θ4εθ40 , (4.80)


for some θ3, θ4 ∈ (0, 1). Note that str1(R) → 0 as p1 → 4/d, hence in the lowercritical case we would need γ0 to be small in [K]0. By the same argument as for(4.67), we obtain

‖γ‖Y0(I) ≤ C(A,B)(εθ30 + εθ40 ) ≪ δ. (4.81)

Next, still assuming (4.78), we have by the exotic Strichartz estimate,

‖γ − γj‖Y(Ij)+ ‖γj+1 − γj‖Y(R) . ‖f ′(w + γ)− f ′(w)‖[Y ]2(Ij) + ‖e‖Y∗(Ij), (4.82)

where the nonlinear difference is estimated by

‖f ′L(w + γ)− f ′

L(w)‖[Y ]2 . ‖(w, γ)‖p2[M ]0‖γ‖[N ]2

+ ‖(w, γ)‖p2[M]2p2

‖γ‖[N ]0

+ ‖(w, γ)‖p2−1[M ]0

‖(w, γ)‖[M]2‖γ‖[N ]0,

(4.83)

where the last term is for p2 > 1 while the second last is for p2 ≤ 1, and similarly

‖f ′S(w + γ)− f ′

S(w)‖[Y ]2 . ‖(w, γ)‖p1[Q]0‖γ‖[P ]2 + ‖(w, γ)‖p1[Q]2p1

‖γ‖[P ]0. (4.84)

Thus we obtain

‖γ − γj‖Y(Ij)+ ‖γj+1 − γj‖Y(R). δp1‖γ‖Y(Ij)

+ ε0, (4.85)

where we used (4.76), (4.78), and the following embeddings in x

[Q]2p1 ⊂ [Q]0, [P ]2 ⊂ [P ]0, [M ]2 + [M ]2p2 ⊂ [M ]0, [N ]2 ⊂ [N ]0. (4.86)

By Lemma 4.3 and Strichartz, we have

‖γ0‖[N ]2(I). ‖γ0‖1−θ5[H]2(I)∩[W ]2(I)

‖γ0‖θ5[M ]0(I).A1−θ5εθ50 ,

‖γ0‖[P ]2(I). ‖γ0‖1−θ6[H]2(I)∩[K]2(I)‖γ0‖θ6[M ]0(I)

.A1−θ6εθ60 ,(4.87)

for some θ5, θ6 ∈ (0, 1). Note that str1(P ) is away from 0 as p1 → 4/d, and so θ5, θ6are uniformly bounded from below. Thus by the same argument as for (4.67),

‖γ‖Y(I) ≤ C(A,B)(εθ50 + εθ60 ) ≪ δ. (4.88)

Hence under the assumption (4.78) we have obtained

‖γ‖[W ]0(I)∩[R]0(I)∩[N ]2(I)∩[P ]2(I).C(A,B)

6∑

k=3

εθk0 ≪ δ. (4.89)

Finally by Strichartz, (4.76) and (4.78), we have

‖u‖Y(Ij). ‖~u(tj)‖L2x+ ‖eq(u) + f ′(u)‖Y∗(Ij)

.A+ ε0 + ‖u‖p1[R]0(Ij)‖u‖[R]2(Ij) + ‖u‖p2[M♯]0(Ij)

‖u‖[W ]2(Ij)

.A+ ε0 + δp1‖u‖Y(Ij).

(4.90)

Hence we obtain

‖u‖Y(Ij).A + ε0, (4.91)

and so

‖u‖Y(I).n(A+ ε0) ≤ C(A,B), (4.92)

which is extended to the full Strichartz norms by Lemma 4.4.


It remains to justify (4.78). By Lemma 4.3(2), we have

‖γ‖[Q]2p1∩[M ]2∩[M ]2p2.∑

k=7,8

‖γ‖1−θk[H]2∩[K]2∩[W ]2‖γ‖θk

[P ]2∩[N ]2, (4.93)

for some θ7, θ8 ∈ (0, 1). If p1 = 4/d, then we need to add [K]0 to the last factor.In either case, by (4.91), (4.76), (4.89), and (4.71), we obtain

‖γ‖([Q]2p1∩[M ]2∩[M ]2p2∩[R]0∩[M♯]0)(Ij)

.C(A,B)εθ0, (4.94)

for some θ ∈ (0, 1). By choosing ε0(A,B) sufficiently small, the last bound can bemade much smaller than δ. Then the assumption (4.78) is justified by continuity int and induction in j. Thus we have obtained the desired estimates. �

5. Profile decomposition

In this section, following Bahouri-Gerard and Kenig-Merle, we investigate be-havior of general sequences of solutions, by asymptotic expansion into a series oftransformation sequences of fixed space-time functions, called profiles. This is thefundamental part for the construction of a critical element in the next section.

5.1. Linear profile decomposition. Here we give the Klein-Gordon version ofBahouri-Gerard’s profile decomposition for the massless free wave equation. Theonly essential difference is that the massive equation does not commute with thescaling transforms, but the proof goes almost the same.

For simple presentation, we introduce the following notation. For any triple(t♦♥, x

♦♥, h

♦♥) ∈ R1+d × (0,∞) with arbitrary suffix ♥ and ♦, let τ♦♥ , T♦

♥ and 〈∇〉♦♥respectively denote the scaled time shift, the unitary and the self-adjoint operatorsin L2(Rd), defined by

τ♦♥ = − t♦♥

h♦♥, T♦

♥ϕ(x) = (h♦♥)−d/2ϕ

(x− x♦♥

h♦♥

), 〈∇〉♦♥ =

√−∆+ (h♦♥)

2. (5.1)

We denote the set of Fourier multipliers on Rd:

MC = {µ = F−1µF | µ ∈ C(Rd), ∃ lim|x|→∞

µ(x) ∈ R}. (5.2)

(practically we need only 1 and |∇|〈∇〉−1 in MC). Also recall the correspondenceu↔ ~u defined in Section 4.1.

Lemma 5.1 (Linear profile decomposition). Let ~vn = ei〈∇〉t~vn(0) be a sequence of

free Klein-Gordon solutions with bounded L2x norm. Then after replacing it with

some subsequence, there exist K ∈ {0, 1, 2 . . . ,∞} and, for each integer j ∈ [0, K),ϕj ∈ L2(Rd) and {(tjn, xjn, hjn)}n∈N ⊂ R×Rd × (0, 1] satisfying the following. Define

~vjn and ~wkn for each j < k ≤ K by

~vjn = ei〈∇〉(t−tjn)T jnϕj , ~vn =

k−1∑

j=0

~vjn + ~wkn, (5.3)

then we have

limk→K

limn→∞

‖~wkn‖L∞t (R;B

−d/2∞,∞ (Rd))

= 0, (5.4)


and for any Fourier multiplier µ ∈ MC, any l < j < k ≤ K and any t ∈ R,

limn→∞

| log(hln/hjn)|+|tln − tjn|+ |xln − xjn|

hln= ∞, (5.5)

limn→∞

〈µ~vln(t)|µ~vjn(t)〉L2x= 0 = lim

n→∞〈µ~vjn(t)|µ~wkn(t)〉L2

x. (5.6)

Moreover, each sequence {hjn}n∈N is either going to 0 or identically 1 for all n.

We call such a sequence {~vjn}n∈N a free concentrating wave for each j, and~wkn the remainder. We say that {(tjn, xjn, hjn)}n and {(tkn, xkn, hkn)} are orthogonal

when (5.5) holds. Note that (5.6) implies

limn→∞

[‖~vn(t)‖2L2

x−∑

j<k

‖~vjn(t)‖2L2x− ‖~wkn‖2L2

x

]= 0. (5.7)

We remark that the case hjn → ∞ is excluded by the presence of the mass, or moreprecisely by the use of inhomogeneous Besov norm for the remainder.

Proof. We introduce a Littlewood-Paley decomposition for the Besov norm. Let

Λ0(x) ∈ S(Rd) such that its Fourier transform Λ0(ξ) = 1 for |ξ| ≤ 1 and Λ0(ξ) = 0for |ξ| ≥ 2. Then we define Λk(x) for any k ∈ N and Λ(0)(x) by the Fourier transforms

Λk(ξ) = Λ0(2−kξ)− Λ0(2

−k+1ξ), Λ(0) = Λ0(ξ)− Λ0(2ξ). (5.8)

Let

ν := limn→∞

‖~vn‖L∞t B

−d/2∞,∞

∼ limn→∞

supt∈R, x∈Rd, k≥0

2−kd/2|Λk ∗ ~vn(t, x)|. (5.9)

If ν = 0, then we are done withK = 0. Otherwise, there exists a sequence (tn, xn, kn)such that for large n

2−knd/2|Λkn ∗ ~vn(tn, xn)| ≥ ν/2. (5.10)

Now we define hn and ψn by

hn = 2−kn, ~vn(tn, x) = Tnψn. (5.11)

Since ψn is bounded in L2x, it converges weakly to some ψ in L2

x, up to an extractionof a subsequence. Moreover,

2−knd/2|Λkn ∗ ~vn(tn, xn)| ={|Λ0 ∗ ψn(0)| (kn = 0)

|Λ(0) ∗ ψn(0)| (kn ≥ 1),(5.12)

and hence by the weak convergence and by Schwarz

‖ψ‖L2x& |〈Λ0|ψ〉|+ |〈Λ(0)|ψ〉| ≥ ν/2. (5.13)

If hn → 0, then we put (t0n, x0n, h

0n) = (tn, xn, hn) and ϕ0 = ψ. Otherwise, we may

assume that hn → ∃h∞ > 0, by extracting a subsequence, and we put

(t0n, x0n, h

0n) = (tn, xn, 1), ϕ0 = h−d/2∞ ψ(x/h∞). (5.14)

Then we have Tnψ − T 0nϕ

0 → 0 strongly in L2x. Now we define ~v0n and ~w1

n by

~v0n = ei〈∇〉(t−t0n)T 0nϕ

0, ~w1n = ~vn − ~v0n. (5.15)


Then (T 0n)

−1 ~w1n(t

0n) = (T 0

n)−1Tnψn−ϕ0 → 0 weakly in L2, and µT 0

n = T 0nµ

0n, where µ

0n

denotes the Fourier multiplier whose symbol is the rescaling of µ’s, that is µ(ξ/h0n).By the definition of MC, the symbol of µ0

n converges including the case h0n → 0, soµ0n → ∃µ0

∞ converges strongly on L2(Rd). Hence

〈µ~v0n(t0n)|µ~w1n(t

0n)〉L2

x= 〈µ0

nϕ0|µ0

n(T0n)

−1 ~w1n(t

0n)〉L2

x→ 0. (5.16)

The left hand side is preserved in t, hence the above holds at any t. This is thedecomposition for k = 1.

Next we apply the above procedure to the sequence ~w1n in place of ~vn. Then either

the Besov norm goes to 0 and K = 1, or otherwise we find the next concentratingwave ~v1n and the remainder ~w2

n, such that for some (t1n, x1n, h

1n) and ϕ

1 ∈ L2(Rd),

~w1n = ~v1n + ~w2

n, ~v1n = ei〈∇〉(t−t1n)T 1nϕ

1, 〈µ~v1n(t)|µ~w2n(t)〉L2

x→ 0, (5.17)

(T 1n)

−1 ~w2n(t

1n) → 0 weakly in L2

x as n→ ∞, and

limn→∞

‖~w1n‖L∞

t B−d/2∞,∞

. ‖ϕ1‖L2. (5.18)

Iterating the above procedure, we obtain the desired decomposition. The L2

orthogonality implies that ‖ϕk‖L2x→ 0 as k → ∞, and then (5.18) (for general k)

gives the decay of the remainder in the Besov norm.It remains to prove the orthogonality (5.5) as well as (5.6). First we have

〈µ~vln(0)|µ~vjn(0)〉 = 〈e−i〈∇〉tlnT lnµlnϕ

l|e−i〈∇〉tjnT jnµjnϕ

j〉 = 〈Sj,ln µlnϕl|µjnϕj〉, (5.19)

where µln = µ(ξ/hln) as before, and Sj,ln is defined by

Sj,ln := (T jn)−1ei〈∇〉(tjn−t

ln)T ln = e−i〈∇〉jnt

j,ln (T jn)

−1T ln = e−i〈∇〉jntj,ln T j,ln , (5.20)

with the sequence

(tj,ln , xj,ln , h

j,ln ) := (tln − tjn, x

ln − xjn, h

ln)/h

jn. (5.21)

Using the last formula in (5.20), (5.5) and uniform time decay of ei〈∇〉jnt : S → S ′,it is easy to observe that Sj,ln → 0 weakly on L2

x as n → ∞ for all j < l. Sinceµln = µ(ξ/hln) and µ

jn are convergent, (5.19) also tends to 0. Then we have also

〈µ~vjn(t)|µ~wkn(t)〉L2x= 〈µ~vjn(t)|µ~wj+1

n (t)−k−1∑

m=j+1

µ~vmn (t)〉L2x→ 0, (5.22)

thus we obtain (5.6). Now suppose that (5.5) fails, then there exists a minimal (l, j)breaking (5.5), with respect to the natural order

(l1, j1) ≤ (l2, j2) ⇐⇒ l1 ≤ l2 and j1 ≤ j2. (5.23)

Then by extracting a subsequence, we may assume that hln → hl∞, log(hln/hjn),

(tln − tjn)/hln and (xln − xjn)/h

ln all converge. Now we inspect

(T ln)−1 ~wl+1

n (tln) =

j∑

m=l+1

Sl,mn ϕm + Sl,jn (T jn)−1 ~wj+1

n (tjn). (5.24)

where Sl,jn converges strongly to a unitary operator, due to the convergence of(tl,jn , x

l,jn , h

l,jn ) and hln. Since Sl,mn → 0 for m < j and (T jn)

−1 ~wj+1n (tjn) → 0 weakly


in L2x, we deduce from the weak limit of (5.24) that ϕk = 0, a contradiction. This

proves the orthogonality (5.5). �

Those free concentrating waves with scaling going to 0 are vanishing in any Besovspace with less regularity. Hence in the subcritical case, we may freeze the scalingto 1 by regarding them as a part of remainder. Hence we have

Corollary 5.2. Let ~vn be a sequence of free Klein-Gordon solutions with bounded L2x

norm. Then after replacing it with some subsequence, there existK ∈ {0, 1, 2 . . . ,∞}and, for each integer j ∈ [0, K), ϕj ∈ L2(Rd) and {(tjn, xjn)}n∈N ⊂ R×Rd satisfying

the following. Define ~vjn and ~wkn for each j < k ≤ K by

~vjn = ei〈∇〉(t−tjn)ϕj(x− xjn), ~vn =k−1∑

j=0

~vjn + ~wkn, (5.25)

then for any s < −d/2, we have

limk→K

limn→∞

‖~wkn‖L∞(R;Bs∞,1(R

d)) = 0, (5.26)

and for any µ ∈ MC, any l < j < k ≤ K and any t ∈ R,

limn→∞

〈µ~vln|µ~vjn〉2L2x= 0 = lim

n→∞〈µ~vjn|µ~wkn〉L2

x, (5.27)

limn→∞

|tjn − tkn|+ |xjn − xkn| = ∞. (5.28)

The orthogonality holds also for the nonlinear energy, which implies that the

decomposition is closed in K+. Recall the vector notation for the energy given inSection 4.1. We will also use the following estimates for 1 < p <∞,

‖[|∇| − 〈∇〉n]ϕ‖Lpx.hn‖〈∇/hn〉−1ϕ‖Lp

x,

‖[|∇|−1 − 〈∇〉−1n ]ϕ‖Lp

x. ‖〈∇/hn〉−2|∇|−1ϕ‖Lp

x,

(5.29)

which hold uniformly for 0 < hn ≤ 1, by Mihlin’s theorem on Fourier multipliers.

Lemma 5.3. Assume that f satisfies (1.36). Let ~vn be a sequence of free Klein-

Gordon solutions satisfying ~vn(0) ∈ K+ and limn→∞ E(~vn(0)) < m. Let ~vn =∑j<k ~v

jn+ ~wkn be the linear profile decomposition given by Lemma 5.1. Except for the

H1 critical case (1.28), it may be given by Lemma 5.2 too. Then we have ~vjn(0) ∈ K+

for large n and all j < K, and

limk→K

limn→∞

∣∣∣E(~vn(0))−∑

j<k

E(~vjn(0))− E(~wkn(0))∣∣∣ = 0. (5.30)

Moreover we have for all j < K

0 ≤ limn→∞

E(~vjn(0)) ≤ limn→∞

E(~vjn(0)) ≤ limn→∞

E(~vn(0)), (5.31)

where the last inequality becomes equality only if K = 1 and ~w1n → 0 in L∞

t L2x.

Proof. First we see that in the exponential case (1.29), all the profiles and remainders

are in the subcritical regime. Since ~vn(0) ∈ K+, Lemma 2.11 implies

‖∇〈∇〉−1ℜ~vn(0)‖2L2x+ ‖ℑ~vn(0)‖2L2

x< 2m ≤ 4π/κ0. (5.32)


For any (θ0, . . . , θk) ∈ C1+k satisfying ‖θ‖L∞ = maxj |θj | ≤ 1, let

vθn =∑

j<k

θjvjn + θkw

kn. (5.33)

Then choosing µ = |∇|〈∇〉−1 ∈ MC in (5.27), we get

limn→∞

supt∈R

‖∇vθn‖2L2x≤ lim

n→∞‖∇〈∇〉−1~vn‖2L2

x=:M < 4π/κ0. (5.34)

Hence there exist κ > κ0 and q ∈ (1, 2) such that qκM < 4π.Now we start proving (5.30) in all the cases. Since the linear version of (5.30) is

given by Lemma 5.1, it suffices to show the orthogonality in F , i.e.

limk→K

limn→∞

∣∣∣F (vn(0))−∑

j<k

F (vjn(0))− F (wkn(0))∣∣∣ = 0. (5.35)

For this we may neglect wkn, because by the decay in B1−d/2∞,∞ and interpolation with

the H1 bound we have

limk→K

limn→∞

‖wkn(0)‖Lpx= 0 (2 < p ≤ 2⋆). (5.36)

In the exponential case, we deal with it as follows. Let v<k+θn = vn − (1 − θ)wkn for0 ≤ θ ≤ 1. Using the Holder and Trudinger-Moser inequalities, we get

|F (vn)− F (v<kn )| ≤∫ 1

0

∫|f ′(v<k+θn )wkn|dxdθ

≤∫ 1

0

dθ‖eqκ|v<k+θn |2 − 1‖1/qL1

x‖wkn‖Lq′

x≤∫ 1

0

dθ

[‖v<k+θn ‖2L2

x

4π − qκM

]1/q‖wkn‖Lq′

x.

(5.37)

In the subcritical/exponential cases, it suffices to have the decay in Bs∞,1 for all

s < 1− d/2, which is given by Lemma 5.2. Thus in any case we may replace vn(0)by v<kn (0) in (5.35).

Next we may discard those j for which τ jn = −tjn/hjn → ±∞, since for anyp ∈ (2, 2⋆] satisfying 1/p = 1/2− s/d with s ∈ (0, 1], we have

‖vjn(0)‖Lpx. ‖e−i〈∇〉jnτ

jn |∇|−sϕj‖Lp

x→ 0 (n→ ∞), (5.38)

by the decay of ei〈∇〉jnt in S → Lp as |t| → ∞, which is uniform in n, and the Sobolev

embedding Hsx ⊂ Lpx.

So extracting a subsequence, we may assume that τ jn → ∃τ j∞ ∈ R for all j. Let

ψj := ℜe−i〈∇〉j∞τ j∞ϕj ∈ L2x(R

d) (5.39)

Then vjn(0)− 〈∇〉−1T jnψj → 0 strongly in H1

x, thus (5.35) has been reduced to

|F (∑

j<k

〈∇〉−1T jnψj)−

∑

j<k

F (〈∇〉−1T jnψj)| → 0. (5.40)

In the subcritical/exponential cases, if hjn → 0 then 〈∇〉−1T jnψj → 0 strongly in

Lpx for 2 ≤ p < 2⋆, so it can be neglected. Hence we may assume that hjn ≡ 1. Theneach T jn〈∇〉−1ψj is getting away from the others as n→ ∞, and so (5.40) follows.

In the critical case, if hjn → 0 then we have by (5.29),

‖〈∇〉−1T jnψj − hjnT

jn|∇|−1ψj‖L2⋆

x. ‖〈∇/hjn〉−2|∇|−1ψj‖L2⋆

x→ 0. (5.41)


Hence we may replace 〈∇〉−1T jnψj in (5.40) by hjnT

jnψ

j for some ψj ∈ L2⋆ , including

the case hjn ≡ 1. Then we may further replace each ψj by

ψjn(x) := ψj(x)×{0 ∃l < j s.t. hln < hjn and (x− xj,ln )/hj,ln ∈ supp ψl,

1 otherwise,(5.42)

where (xj,ln , hj,ln ) is defined in (5.21), because (5.5) after the above reduction implies

either hj,ln → 0 or |xj,ln | → ∞, and so ψj → ψj at almost every x ∈ Rd as n → ∞,and strongly in L2⋆

x by the dominated convergence theorem. Now the decompositionis trivial

F (∑

j<k

hjnTjnψ

jn) =

∑

j<k

F (hjnTjnψ

jn), (5.43)

by the support property of ψjn. Thus we have obtained (5.35) and (5.30).By exactly the same argument, we obtain also

limk→K

limn→∞

∣∣∣Kα,β(~vn(0))−∑

j<k

Kα,β(~vjn(0))− Kα,β(~w

kn(0))

∣∣∣ = 0. (5.44)

The remaining conclusions follow from the next lemma. �

Lemma 5.4 (Decomposition in K+). Assume that f satisfies (1.36). Let k ∈ N and

ϕ0, . . . , ϕk ∈ H1(Rd). Assume that

E(k∑

j=0

ϕj) ≤ m− δ, Kα,β(k∑

j=0

ϕj) ≥ −ε,

E(k∑

j=0

ϕj) ≥k∑

j=0

E(ϕj)− ε, Kα,β(k∑

j=0

ϕj) ≤k∑

j=0

Kα,β(ϕj) + ε,

(5.45)

for some (α, β) in (1.16) and some δ, ε > 0 satisfying ε(1+2/µ) < δ. Then ϕj ∈ K+

for all j = 0, . . . , k, i.e. 0 ≤ E(ϕj) < m and Kα,β(ϕj) ≥ 0 for all (α, β) in (1.16).

Proof. Let ψj = ℜ〈∇〉−1ϕj and suppose that K(ϕl) < 0 for some l. Then K(ψl) ≤K(ϕl) < 0 and so H(ψl) ≥ m. Since H is non-negative,

m ≤k∑

j=0

H(ψj) ≤k∑

j=0

[H(ψj) +HQ(ℑ〈∇〉−1ϕj)] =k∑

j=0

[E(ϕj)− K(ϕj)/µ]

≤ E(

k∑

j=0

ϕj)− K(

k∑

j=0

ϕj)/µ+ ε(1 + 1/µ) < m,

(5.46)

where HQ denotes the quadratic part of H . Hence K(ψj) ≥ 0 for all j, and so

E(ϕj) ≥ J(ψj) = H(ψj) +K(ψj)/µ ≥ 0. �


5.2. Nonlinear profile decomposition. The next step is to construct a similardecomposition for the nonlinear solutions with the same initial data.

First we construct a nonlinear profile corresponding to a free concentrating wave.Let ~vn be a free concentrating wave for a sequence (tn, xn, hn) ∈ R× Rd × (0, 1],

(i∂t + 〈∇〉)~vn = 0, ~vn(tn) = Tnψ, ψ(x) ∈ L2, (5.47)

satisfying ~vn(0) ∈ K+. Here we use Lemma 5.1 only in the H1 critical case, andLemma 5.2 in the subcritical/exponential cases. Hence hn → 0 can happen only inthe critical case, otherwise hn ≡ 1. Let un be the nonlinear solution with the sameinitial data

(i∂t + 〈∇〉)~un = f ′(un), ~un(0) = ~vn(0) ∈ K+, (5.48)

which may be local in time. Next we define ~Vn and ~Un by undoing the transforms

~vn = Tn~Vn((t− tn)/hn), ~un = Tn~Un((t− tn)/hn). (5.49)

Then they satisfy the rescaled equations

~Vn = eit〈∇〉nψ, ~Un = ~Vn − i

∫ t

τn

ei(t−s)〈∇〉nf ′(ℜ〈∇〉−1n~Un)ds, (5.50)

where τn = −tn/hn. Extracting a subsequence, we may assume convergence

hn → ∃h∞ ∈ [0, 1], τn → ∃τ∞ ∈ [−∞,∞]. (5.51)

Then the limit equations are naturally given by

~V∞ = eit〈∇〉∞ψ, ~U∞ = ~V∞ − i

∫ t

τ∞

ei(t−s)〈∇〉∞f ′(U∞)ds, (5.52)

where U∞ is defined by

U∞ := ℜ〈∇〉−1∞~U∞ =

{ℜ〈∇〉−1~U∞ (h∞ = 1),

ℜ|∇|−1~U∞ (h∞ = 0).(5.53)

The unique existence of a local solution ~U∞ around t = τ∞ is known in all cases,including h∞ = 0 and τ∞ = ±∞ (the latter corresponding to the existence of thewave operators), by using the standard iteration with the Strichartz estimate. In the

exponential case, it requires that ~U∞ is in the subcritical regime in the Trudinger-

Moser inequality. It is guaranteed by Lemma 5.3, because ~V∞(t) ∈ K+ for t close to

τ∞, and so ~U∞(t) ∈ K+ for all t in its existence interval.~U∞ on the maximal existence interval is called the nonlinear profile associated

with the free concentrating wave ~vn. The nonlinear concentrating wave ~u(n)associated with ~vn is defined by

~u(n) = Tn~U∞((t− tn)/hn). (5.54)

If h∞ = 1 then u(n) solves NLKG. If h∞ = 0 then it solves

(∂2t −∆+ 1)u(n) = (i∂t + 〈∇〉)~u(n)= (〈∇〉 − |∇|)~u(n) + f ′(|∇|−1〈∇〉u(n)).

(5.55)


The existence time of u(n) may be finite and even go to 0, but at least we have

‖~un(0)− ~u(n)(0)‖L2x= ‖~Vn(τn)− ~U∞(τn)‖L2

x

≤ ‖~Vn(τn)− ~V∞(τn)‖L2x+ ‖~V∞(τn)− ~U∞(τn)‖L2

x→ 0.

(5.56)

Let un be a sequence of (local) solutions of NLKG in K+ around t = 0, and letvn be the sequence of the free solutions with the same initial data. We consider thelinear profile decomposition given by Lemma 5.1 or 5.2,

~vn =k−1∑

j=0

~vjn + ~wkn, ~vjn = ei〈∇〉(t−tjn)T jnϕj. (5.57)

With each free concentrating wave {~vjn}n∈N, we associate the nonlinear concentratingwave {~uj(n)}n∈N. A nonlinear profile decomposition of un is given by

~u<k(n) :=k−1∑

j=0

~uj(n). (5.58)

We are going to prove that ~u<k(n) is a good approximation for ~un, provided that each

nonlinear profile has finite global Strichartz norm (in Lemma 5.6). Now we definethe Strichartz norms for the profile decomposition, using the notation in Section 4.2.Let ST and ST ∗ be the function spaces on R1+d defined by

ST = [W ]2 ∩ [K]2, ST ∗ = [W ∗(1)]2 + [K∗(1)]2 + L1tL

2x, (5.59)

where the exponents W and K as well as their duals are as defined in (4.24) and(4.9). The Strichartz norm for the nonlinear profile depends on the scaling h♦∞ forany suffix ♦;

ST♦∞ :=

{[W ]2 ∩ [K]2 (h♦∞ = 1),

[W ]•2 (h♦∞ = 0).(5.60)

In other words, we take the scaling invariant part if h♦n → +0, which can happenonly in the H1 critical case. The following estimate is convenient to treat theconcentrating case: For any S ∈ [0, 1]× [0, 1/2]× [0, 1] we have

‖u(n)‖[S]2(R) . (hn)1−reg0(S)‖U∞‖[S]•2(R), (5.61)

where U∞ is as defined in (5.53). Indeed, using B0p,2 ⊂ Lp with p = 1/S2 ≥ 2 in the

lower frequencies, we have

‖u(n)‖[S]2 . ‖|∇|−S3〈∇〉S3u(n)‖[S]•2∼ (hn)

1−reg0(S)‖ℜ|∇|−S3〈∇〉S3−1n

~U j∞‖[S]•2 . (hn)

1−reg0(S)‖U j∞‖[S]•2 .

(5.62)

Concerning the orthogonality in the Strichartz norms, we have

Lemma 5.5. Assume that f satisfies (1.36). Suppose that in the nonlinear profile

decomposition (5.58) we have

‖U j∞‖ST j

∞(R) + ‖~U j∞‖L∞

t L2x(R) <∞ (5.63)


for each j < K. Then we have for any finite interval I, any j < K and any k ≤ K,

limn→∞

‖uj(n)‖ST (I). ‖U j∞‖ST j

∞(R), (5.64)

limn→∞

‖u<k(n)‖2ST (I). limn→∞

∑

j<k

‖uj(n)‖2ST (I), (5.65)

where the implicit constants do not depend on I, j or k. We have also

limn→∞

‖f ′(u<k(n))−∑

j<k

f ′((〈∇〉j∞)−1〈∇〉uj(n))‖ST ∗(I) = 0. (5.66)

Proof. First note that if hj∞ = 1 then uj(n) is just a sequence of space-time translations

of U j∞. In particular, (5.64) is trivial in that case.

Next we prove (5.64) in the case hj∞ = 0, which is only in the H1 critical case.For the moment we drop the superscript j. For the [W ]2 part, (5.61) gives us

‖u(n)‖[W ]2(I). ‖U∞‖[W ]•2(R)= ‖U∞‖ST j

∞(R). (5.67)

For the [K]2 part, let V be the following interpolation between H and W

V :=1

d+ 2H +

d+ 1

d+ 2W = K +

(−1, 0, 1)

2(d+ 2). (5.68)

Then using Holder in t and (5.61) together with reg0(K) = (d+ 1)/(d+ 2), we get

‖u(n)‖[K]2(I). ‖u(n)‖[V 12 ]2(I)

|I| 12(d+2) . (hn)

12(d+2)‖U∞‖[V ]•2(R)

|I| 12(d+2) → 0, (5.69)

as n→ ∞. Thus we have proved (5.64).

Next we prove (5.65) in the subcritical/exponential cases. Define U j∞,R, u

j(n),R for

R ≫ 1 and u<k(n),R by

U j∞,R = χR(t, x)U

j∞, uj(n),R = T jnU

j∞,R(t− tjn), u<k(n),R =

∑

j<k

uj(n),R, (5.70)

where χR is the cut-off defined in (1.23). Then we have

‖u<k(n) − u<k(n),R‖ST (R) ≤∑

j<k

‖(1− χR(t, x))Uj∞‖ST (R) → 0, (R → +0) (5.71)

so we may replace u<k(n) by u<k(n),R. Let δ

lm denote the difference operator

δlmϕ(x) = ϕ(x− 2−mel)− ϕ(x), (5.72)

where el denotes the l-th unit vector in Rd. Each Besov norm in ST is equivalent to

d∑

l=1

∥∥∥∑

j<k

2smδlmuj(n),R

∥∥∥Lpt ℓ

2m≥0L

qx

+ ‖∑

j<k

uj(n),R‖LptL

qx, (5.73)

where (1/p, 1/q, s) =W or K. (5.28) implies that each supp uj(n),R is away from the

others at least by distance 2 for large n, and then supp δlmuj(n),R are also disjoint for

j < k at each l, m. Hence the first norm in (5.73) equals

‖2smδlmuj(n),R‖Lpt ℓ

2m≥0L

qxℓ2j<k

≤ ‖2smδlmuj(n),R‖ℓ2j<kLpt ℓ

2m≥0L

qx. ‖uj(n),R‖ℓ2j<kL

ptB

sq,2, (5.74)


where the first inequality is by Minkowski. Thus we have obtained (5.65) in thesubcritical/exponential cases.

Next we prove (5.65) in the H1 critical case. For the nonlinear concentratingwaves with hj∞ = 1, the above argument works. For those with hj∞ = 0, the Kcomponent is vanishing by (5.69). Hence it suffices to estimate [W ]2 in the case allhjn tend to 0 as j → ∞. Using that W3 = 1/2 ∈ (0, 1), we have

‖u<k(n)‖[W ]2(R). ‖|∇|−1〈∇〉u<k(n)‖[W ]•2(R)= ‖ℜ|∇|−1~u<k(n)‖[W ]•2(R)

∼ ‖∑

j<k

uj,ln,m‖Lpt ℓ

2m∈Z

Lqx, (5.75)

where we put (1/p, 1/q, s) = W and

uj,ln,m := 2smδlmhjnT

jnU

j∞((t− tjn)/h

jn), (5.76)

where δlm is the difference operator defined in (5.72). For R ≫ 1, let

uj,ln,m,R(t, x) :=

{χhjnR(t− tjn, x− xjn)u

j,ln,m(t, x) (|m− log2 h

jn| ≤ R)

0 (|m− log2 hjn| > R),

(5.77)

where χ∗ is as in (1.23). Then by the same computation as for (5.61), we have

‖uj,ln,m − uj,ln,m,R‖Lpt ℓ

2m∈Z

Lqx. ‖2smδlmU j

∞‖Lpt ℓ

2mL

qx(|t|+|m|+|x|>R) → 0, (5.78)

as R → ∞ uniformly in n. Hence we may replace uj,ln,m by uj,ln,m,R in (5.75). The

orthogonality (5.5) implies that {supp(t,m,x) uj,ln,m,R}j<k becomes mutually disjoint for

large n. Then arguing as in (5.74), we obtain (5.65).To prove (5.66) in the subcritical/exponential cases is easier than (5.65), because

after the smooth cut-off, we have for large n

f ′(u<k(n),R) =∑

j<k

f ′(uj(n),R). (5.79)

Note that the uj(n) ∈ ST implies that the full Strichartz norms are finite by Lemma

4.4. The error for f ′(u<k(n)) coming from the cut-off is small in ST ∗ by (4.61)–(4.64)

if d ≤ 4. When d ≥ 5, the difference estimates in the proof of Lemma 4.5 arenot sufficient because they control only the exotic norm Y . In order to estimatethe difference in the admissbile dual norm ST ∗(I), we introduce the following newexponents:

Hε := (ε2,1− ε

2, 0), Wε := W − p2ε(d,−1, 0), M ♯

ε :=M ♯ + ε(d,−1, 0), (5.80)

where W and M ♯ were defined in (4.24) and (4.31), and ε ∈ (0, p1) is fixed smallenough to have

str0(Hε), str0(M ♯

ε), str0(Wε) < 0, reg0(Hε) < 1,

reg0(Wε) = reg0(W ) = 1, reg0(M ♯ε) = reg0(M ♯) ≤ 1,

Wε + p2M♯ε =W + p2M

♯ = W ∗(1).

(5.81)

Then we have for any u and v,

‖f ′S(u)− f ′

S(v)‖L1tL

2x(I)

. |I|1−ε2‖u− v‖[Hε]0(I)(‖u‖L∞t L2

x(I)+ ‖v‖L∞

t L2x(I)

)ε, (5.82)


because |f ′S(u)− f ′

S(v)|. |u− v|(|u|+ |v|)ε. For large u, we have if p2 ≥ 1,

‖f ′L(u)− f ′

L(v)‖[W ∗(1)]2

. ‖u‖p2[M♯

ε]0‖u− v‖[Wε]2 + ‖u− v‖[M♯

ε ]0(‖u‖[M♯

ε]0+ ‖v‖[M♯

ε]0)p2−1‖v‖[Wε]2,

(5.83)

and if p2 < 1,

‖f ′L(u)− f ′

L(v)‖[W ∗(1)ε ]2

. ‖u‖p2[M♯

ε ]0‖u− v‖[Wε]2 + ‖u− v‖p2

[M♯ε]0‖v‖[Wε]2. (5.84)

The latter estimate is not Lipschitz in u − v, but sufficient for our purpose here.4

Thus we obtain (5.66) in the subcritical/exponential cases.It remains to prove (5.66) in the H1 critical case, where we need further cut-off

to get a disjoint sum. First we see that each uj(n) in u<k(n) may be replaced with

uj〈n〉 := (〈∇〉j∞)−1〈∇〉uj(n) = hjnTjnU

j∞((t− tjn)/h

jn). (5.85)

For the moment we drop the superscript j. Let p2 = 4/(d−2) and h∞ = 0. If d ≤ 4,then we have by using (4.62) and (5.29)

‖f ′(u(n))− f ′(u〈n〉)‖L1tL

2x(R)

. ‖u〈n〉‖p2[L]0(R)‖u(n) − u〈n〉‖[L]0(R)∼ ‖U∞‖p2[L]0(R)‖[|∇|〈∇〉−1

n − 1]U∞‖[L]0(R). ‖U∞‖p2[L]0(R)‖〈∇/hn〉

−2U∞‖[L]0(R) → 0,

(5.86)

since U∞ ∈ [H ]•2 ∩ [W ]•2 ⊂ [L]0 by the homogeneous version of Lemma 4.3(1).If d ≥ 5, we introduce a new exponent

G :=d− 2

d+ 2

(1

d+ 1,d+ 3

2(d+ 1), 0

). (5.87)

Then reg0(G) = 1, str0(G) < 0 and

(2⋆ − 1)G =W ∗(1) − (1, 0, 1)

2. (5.88)

Hence

‖f ′(u(n))− f ′(u〈n〉)‖[W ∗(1)]2(I). ‖f ′(u(n))− f ′(u〈n〉)‖[W ∗(1)]•2(R)

+ |I|1/2‖f ′(u(n))− f ′(u〈n〉)‖[(2⋆−1)G]0(I),(5.89)

where the first term on the right is dominated by (the homogeneous version of(5.83)–(5.84))

‖u〈n〉‖p2[M♯ε ]0(R)

‖u(n) − u〈n〉‖[Wε]•2(R)+ ‖u(n) − u〈n〉‖θ[M♯

ε ]0(R)‖(u〈n〉, u(n))‖p2−θ[Wε]•2(R)

. ‖U∞‖p2[M♯

ε ]0(R)‖〈∇/hn〉−2U∞‖[Wε]•2(R)

+ ‖〈∇/hn〉−2U∞‖θ[M♯

ε]0(R)‖U∞‖p2−θ[Wε]•2(R)

,(5.90)

where θ := min(p2, 1). The right hand side goes to 0, since U∞ ∈ [H ]•2∩[Wε]•2 ⊂ [M ♯

ε ]0by the homogeneous version of Lemma 4.3(1). Similarly, the last term in (5.89) isbounded by

‖u〈n〉‖p2[G]0(R)‖u(n) − u〈n〉‖[G]0(R) ∼ ‖U∞‖p2[G]0(R)

‖〈∇/hn〉−2U∞‖[G]0(R) → 0. (5.91)

4The situation is different from the long-time iteration in the previous section, where we neededthe exotic Strichartz estimate in order to get the Lipschitz estimate for the iteration along thenumerous time intervals.


Thus it suffices to show

‖f ′(∑

j<k

uj〈n〉)−∑

j<k

f ′(uj〈n〉)‖ST ∗(I) → 0. (5.92)

Now we define U jn,R for any R ≫ 1 by

U jn,R(t, x) = χR(t, x)U

j∞(t, x)

×∏

{(1− χhj,ln R)(t− tj,ln , x− xj,ln ) | 1 ≤ l < k, hlnR < hjn},(5.93)

where χR and (tj,ln , xj,ln , h

j,ln ) are as defined respectively in (1.23) and (5.21). Then

it is uniformly bounded in [H ]•2(R) ∩ [W ]•2(R), and U jn,R → χRU

j∞ in [M ♯]0(R) as

n → ∞, because either hj,ln → 0 or |tj,ln | + |xj,ln | → ∞ by the orthogonality (5.5).Then by the homogeneous version of Lemma 4.3(2), it converges also in [L]0(R) (if

d ≤ 4), [Wε]•2(R) and [M ♯

ε ]0(R). Moreover, we have χRUj∞ → U j

∞ as R → ∞ in thesame spaces.

Hence we may replace uj〈n〉 by uj〈n〉,R := hjnT

jnU

jn,R((t − tjn)/h

jn), and then we get

the desired result, since {supp(t,x) uj〈n〉,R}j<k are mutually disjoint for large n, and so

f ′(∑

j<k

uj〈n〉,R) =∑

j<k

f ′(uj〈n〉,R), (5.94)

which concludes the proof of (5.66). �

The next lemma is the conclusion of this section.

Lemma 5.6. Assume that f satisfies (1.36). Let un be a sequence of local solutions

of NLKG around t = 0 in K+ satisfying limn→∞E(un) < m. Suppose that in its

nonlinear profile decomposition (5.58), every nonlinear profile ~U j∞ has finite global

Strichartz and energy norms, i.e.

‖U j∞‖ST j

∞(R) + ‖~U j∞‖L∞

t L2x(R) <∞. (5.95)

Then un is bounded for large n in the Strichartz and the energy norms, i.e.

limn→∞

‖un‖ST (R) + ‖~un‖L∞t L2

x(R) <∞. (5.96)

Proof. We will apply the perturbation lemma to u<k(n)+wkn as an approximate solution.

First observe that

‖~un(0)− ~u<k(n)(0)− wkn(0)‖L2x≤∑

j<k

‖~vjn(0)− ~uj(n)(0)‖L2x= o(1), (5.97)

and

‖~un(0)‖2L2 = ‖~vn‖2L2x≥∑

j<k

‖~vjn‖2L2x+ o(1) =

∑

j<k

‖~uj(n)(0)‖2L2x+ o(1), (5.98)

where o(1) → 0 as n → ∞. Hence except for a finite set J ⊂ N, the energy of uj(n)with j 6∈ J is smaller than the iteration threshold, which implies

‖uj(n)‖ST (R). ‖~uj(n)(0)‖L2x

(j 6∈ J). (5.99)


Combining (5.65), (5.64), (5.99) and (5.98), we obtain for any finite interval I,

supk

limn→∞

‖u<k(n)‖2ST (I).∑

j∈J

‖U j∞‖2

ST j∞+ lim

n→∞‖~un(0)‖2L2

x<∞. (5.100)

The equation of u<k(n) is given by

eq(u<k(n)) =∑

j<k

(〈∇〉 − 〈∇〉j∞)~uj(n) + f ′(u<k(n))−∑

j<k

f ′(uj〈n〉), (5.101)

where uj〈n〉 = (〈∇〉j∞)−1〈∇〉uj(n) as before. The nonlinear part goes to 0 by (5.66),

while the linear part vanishes if hj∞ = 1, and is dominated if hj∞ = 0 by

‖(〈∇〉 − |∇|)~uj(n)‖L1tL

2x(I)

. |I|‖〈∇〉−1~uj(n)‖L∞t L2

x(R)

∼ |I|‖〈∇/hjn〉−1~U j∞‖L∞

t L2x(R) → 0 (n→ ∞),

(5.102)

by continuity in t for bounded t, and by the scattering of U j∞ for |t| → ∞, which

follows from ‖U j∞‖[W ]•2(R)

<∞. Hence Lemma 4.4 gives for any 1-admissible Z

supk

limn→∞

‖u<k(n)‖[Z]2(R) <∞. (5.103)

On the other hand, by Lemma 4.3 we can extend the smallness of wkn fromL∞t B

s∞,∞ to the other spaces that we need for the nonlinear difference estimates,

i.e. [S]0, [L]0, [X ]2, [Hε]0, [M♯ε ]0, and [Wε]2, depending on d and f . In addition, in

the exponential case (1.29), Lemmas 5.3 and 2.11 imply that u<k(n) and wkn are both

in the subcritical regime for the Trudinger-Moser inequality. Putting them togetherwith the above bounds on u<k(n) in the nonlinear difference estimates (4.61)–(4.64) or

(5.82)–(5.84), we get

limk→K

limn→∞

‖f ′(u<k(n) + wkn)− f ′(u<k(n))‖ST ∗(I) = 0, (5.104)

and so

limk→K

limn→∞

‖eq(u<k(n) + wkn)‖ST ∗(I) = 0. (5.105)

Hence for k sufficiently close to K and n large enough, the true solution un andthe approximate solution u<k(n) + wkn satisfy all the assumptions of the perturbation

Lemma 4.5. Hence un is bounded in global Strichartz norms for large n. �

6. Extraction of a critical element

In this section, we prove that if uniform global Strichartz bound fails strictlybelow the variational threshold m, then we have a global solution in K+ with infiniteStrichartz norm and with the minimal energy, which is called a critical element.

Let E⋆ be the threshold for the uniform Strichartz bound. More precisely,

E⋆ := sup{A > 0 | S(A) <∞}, (6.1)

where S(A) denotes the supremum of ‖u‖ST (I) for any strong solution u in K+ onany interval I satisfying E(u) ≤ A.

The small energy scattering tells us E⋆ > 0, and the presence of the groundstate tells us E⋆ ≤ m, at least in the subcritical case, and also in the other cases ifwe allow complex-valued solutions, because the stationary solutions with different


masses yield standing wave solutions of the original NLKG. Anyway, we are goingto prove E⋆ ≥ m by contradiction.

We remark that there is an alternative threshold:

E⋆FS := sup

{A > 0

∣∣∣∣∣If u is a solution in K+ of NLKG

with E(u) ≤ A, then ‖u‖ST (R) <∞

}. (6.2)

Obviously E⋆ ≤ E⋆FS. Kenig-Merle [26] chose this definition. The advantage of using

E⋆ is that E⋆ ≥ m implies uniform bound on the global Strichartz norms below m,which is very important in applications where we want to perturb the equation.

The next lemma is the conclusion of this section.

Lemma 6.1. Assume that f satisfies (1.36), and let un be a sequence of solutions

of NLKG in K+ on In ⊂ R satisfying

E(un) → E⋆ < m, ‖un‖ST (In) → ∞ (n→ ∞). (6.3)

Then there exists a global solution u∗ of NLKG in K+ satisfying

E(u∗) = E⋆, ‖u∗‖ST (R) = ∞. (6.4)

In addition, there are a sequence (tn, xn) ∈ R× Rd and ϕ ∈ L2(Rd) such that along

some subsequence,

‖~un(0, x)− e−i〈∇〉tnϕ(x− xn)‖L2x→ 0. (6.5)

We call such a global solution u∗ a critical element. Observe that by thedefinition of E⋆, we can find such a sequence un, once we have E⋆ < m.

Proof. We can translate un in t so that 0 ∈ In for all n. Then we consider the linearand nonlinear profile decompositions of un, using Lemma 5.1 in the H1 critical case(1.28) and Lemma 5.2 in the subcritical/exponential cases.

ei〈∇〉t~un(0) =∑

j<k

~vjn + ~wkn, ~vjn = ei〈∇〉(t−tjn)T jnϕj ,

u<k(n) =∑

j<k

uj(n), ~uj(n) = T jn~U j∞((t− tjn)/h

jn),

‖~vjn(0)− ~uj(n)(0)‖L2x→ 0 (n→ ∞).

(6.6)

Lemma 5.6 precludes that all the nonlinear profiles ~U j∞ have finite global Strichartz

norm. On the other hand, every solution of NLKG in K+ with energy less than E⋆

has global finite Strichartz norm by the definition of E⋆. Hence by Lemma 5.3 wededuce that there is only one profile i.e. K = 1, and moreover

E(~u0(n)) = E⋆, ~u0(n)(0) ∈ K+, ‖U0∞‖ST 0

∞(R) = ∞, limn→∞

‖~w1n‖L∞

t L2x= 0. (6.7)

If h0n → 0 in the critical case, then U0∞ = |∇|−1ℜ~U0

∞ solves the massless equation

(∂2t −∆)U0∞ = f ′(U0

∞), (6.8)

and satisfies

E0(U0∞) = E⋆ < m = J (0)(Q), Kw(U0

∞(0)) ≥ 0, ‖U0∞‖[W ]•2

= ∞, (6.9)


where Q is the massless ground state and Kw is the massless version of K. However,Kenig-Merle [26] has proven that there is no such solution.5 Hence h0n ≡ 1 in allcases, and we obtain (6.5).

Hence h0n ≡ 1 in all cases, and we obtain (6.5).

It remains to prove that U0∞ = 〈∇〉−1ℜ~U0

∞ is a global solution. Suppose not. Thenwe can choose a sequence tn ∈ R approaching the maximal existence time. Since

the sequence of solutions U0∞(t+ tn) satisfies the assumption of this lemma, we may

apply the above argument to it. In particular, from (6.5) we obtain

‖~U0∞(tn)− e−i〈∇〉t′nψ(x− x′n)‖L2

x→ 0, (6.10)

for some ψ ∈ L2x and another sequence (t′n, x

′n) ∈ R× Rd. Let ~v := ei〈∇〉tψ. Since it

is a free solution, for any ε > 0 there is δ > 0 such that for any interval I satisfying|I| ≤ 2δ, we have ‖〈∇〉−1~v‖ST (I) ≤ ε, where ST = [W ]2 ∩ [K]2 as in (5.59). Then(6.10) implies that

limn→∞

‖〈∇〉−1ei〈∇〉t~U0∞(tn)‖ST (−δ,δ) ≤ ε. (6.11)

If ε > 0 is small enough, this implies that the solution U0∞ exists on (tn − δ, tn + δ),

by the iteration argument, for large n. This contradicts the choice of tn. Hence U0∞

is global and so a critical element. �

7. Extinction of the critical element

In this section, we prove that the critical element can not exist by deriving acontradiction from a few properties of it. The main idea follows [26, 27]. Let ucbe a critical element given by Lemma 6.1. Since NLKG is symmetric in t, we mayassume that ‖uc‖ST (0,∞) = ∞. We call such u a forward critical element. Notethat since the critical element is in K+, we have EQ(u; t) ∼ E(u) uniformly, byLemma 2.10.

7.1. Compactness. First we show that the trajectory of a forward critical elementis precompact for positive time in the energy space modulo spatial translations.

Lemma 7.1. Assume that f satisfies (1.36), and let uc be a forward critical element.

Then there exists c : (0,∞) → Rd such that the set

{(u, u)(t, x− c(t)) | 0 < t <∞} (7.1)

is precompact in H1(Rd)× L2(Rd).

Proof. The proof of Kenig-Merle [26] can be adapted verbatim, but we give a sketchfor the sake of completeness. Recall the convention u↔ ~u defined in Section 4.1.

It suffices to prove precompactness of {~u(tn)} in L2x for any t1, t2, · · · > 0. If tn

converges, then it is trivial from the continuity in t. Hence we may assume thattn → ∞. Applying Lemma 6.1 to the sequence of solutions u(t+ tn), we get anothersequence (t′n, x

′n) ∈ R1+d and ϕ ∈ L2 such that

~u(tn, x)− e−i〈∇〉t′nϕ(x− x′n) → 0 in L2x (n→ ∞). (7.2)

5[26] is restricted to the dimensions d ≤ 5 for simplicity of the perturbation argument, but theelimination of critical elements works in any higher dimensions.


If t′n → −∞, then we have

‖ei〈∇〉t~u(tn)‖ST (0,∞) = ‖ei〈∇〉tϕ‖ST (−t′n,∞) + o(1) → 0, (7.3)

so that we can solve NLKG of u for t > tn with large n globally by iteration withsmall Strichartz norms, contradicting its forward criticality.

If t′n → +∞, then we have

‖ei〈∇〉t~u(tn)‖ST (−∞,0) = ‖ei〈∇〉tϕ‖ST (−∞,−t′n) + o(1) → 0, (7.4)

so that we can solve NLKG of u for t < tn with large n with diminishing Strichartznorms, which implies u = 0 by taking the limit, a contradiction.

Thus t′n is precompact, so is ~u(tn, x+ x′n) in L2x by (7.2). �

As a consequence, the energy of u stays within a fixed radius for all positive time,modulo arbitrarily small rest. More precisely, we define the exterior energy by

ER,c(u; t) =

∫

|x−c|≥R

|ut|2 + |∇u|2 + |u|2 + |f(u)|+ |uf ′(u)|dx, (7.5)

for any R > 0 and c ∈ Rd. Then we have

Corollary 7.2. Let u be a forward critical element. Then for any ε > 0, there exist

R0(ε) > 0 and c(t) : (0,∞) → Rd such that at any t > 0 we have

ER0,c(t)(u; t) ≤ εE(u). (7.6)

7.2. Zero momentum and non-propagation. Next we observe that the criticalelement can not move with any positive speed in the sense of energy. For that wefirst need to see that the (conserved) momentum

P (u) :=

∫

Rd

ut∇udx ∈ Rd (7.7)

is zero for any critical element u.

Lemma 7.3. For any critical element u, we have P (u) = 0.

Proof. For j = 1, . . . , d and λ ∈ R, we define the operator Lλj of Lorentz transform

Lλj u(x0, . . . , xd) = u(y0, . . . , yd),

y0 = x0 coshλ+ xj sinhλ, yj = x0 sinh λ+ xj coshλ, yk = xk (k 6= 0, j),(7.8)

then we have Lαj Lβj = Lα+βj . Since ∂λy0 = yj and ∂λyj = y0, we have

∂λLλju = Lλj [(xj∂t + t∂j)u]. (7.9)

Also we have

∂tLλj = Lλj (s∂t + c∂j), ∂ttL

λj = Lλj (s

2∂tt + 2sc∂tj + c2∂jj),

∂jLλj = Lλj (c∂t + s∂j), ∂jjL

λj = Lλj (c

2∂tt + 2sc∂tj + s2∂jj),(7.10)

where s := sinhλ and c := coshλ. In particular [∂2t − ∆, Lλj ] = 0, and so Lλj mapsglobal solutions to themselves. For the space-time norm, we have

∫∫Lλj vdtdxj =

∫∫v

∣∣∣∣(c ss c

)∣∣∣∣ dtdxj =∫∫

vdtdxj, (7.11)


hence Lλj preserves all Lpt,x(R1+d) norm. For any solution u, we have

∂0λE(Lλj u) = 〈ut|∂0λ∂tLλj u〉+ 〈∇u|∂0λ∇Lλj u〉+ 〈u− f ′(u)|∂0λLλj u〉

= 〈ut|xjutt + tutj + uj〉+ 〈uk|xjukt + tukj + δkjut〉+ 〈u− f ′(u)|xjut + tuj〉

= 〈xjut|∆u〉+ 2〈ut|uj〉 − 〈xjukt|uk〉 = 〈ut, uj〉 = P (u),

(7.12)

where ∂0λ := ∂λ|λ=0. If Pj(u) 6= 0 for some j, then we obtain another global solutionLλj u, which has smaller energy and infinite Strichartz norm. It also belongs to K+,

by continuity. More precisely, the continuity of Lλj u in λ in the energy space easilyfollows from the local wellposedness if u has compactly supported initial data. Thenthe original solution is approximated by smooth cut-off using the finite propagationproperty. Thus we obtain another critical element with less energy, a contradiction.Hence P (u) = 0. �

Next we see stillness of critical elements in terms of the energy propagation. Forany R > 0, we define the localized center of energy XR(t) ∈ Rd by

XR(u; t) :=

∫χR(x)xe(u)(t, x)dx, (7.13)

where χR is as defined in (1.23), and e(u) denote the energy density of u, namely

e(u) = (|ut|2 + |∇u|2 + |u|2)/2− f(u). (7.14)

From the energy identity e(u) = ∇ · (ut∇u), we get for any solution u

d

dtXR(u; t) = −dP (u) +

∫[d(1− χR(x)) + (r∂r)χR(x)]ut∇u. (7.15)

If u is a critical element, the first term disappears by the above lemma, so we have∣∣∣∣d

dtXR(u; t)

∣∣∣∣ .ER,0(u; t). (7.16)

Moreover, since u is in K+, by Lemma 2.12 there exists δ0 ∈ (0, 1) such that

K1,0(u(t)) ≥ δ0‖u(t)‖2H1 (7.17)

for all t ∈ R.

Lemma 7.4. Let u be a forward critical element, and let R0(ε) > 0, c(t) ∈ Rd and

δ0 > 0 be as in (7.6) and (7.17). If 0 < ε≪ δ0 and R ≫ R0(ε) then we have

|c(t)− c(0)| ≤ R− R0(ε), (7.18)

for 0 < t < t0 till some t0& δ0R/ε.

Proof. By translation in x, we may assume that c(0) = 0. Let t0 be the final timefor the above property

t0 = inf{t > 0 | |c(t)| ≥ R− R0}. (7.19)


Then the finite speed of propagation implies that t0 > 0. For any 0 < t < t0 wehave |c(t)| ≤ R−R0, hence by (7.6) we have ER,0 ≤ εE(u), and so by (7.16) we get

∣∣∣∣d

dtXR(u; t)

∣∣∣∣ . εE(u). (7.20)

Next we expand it around c:

c(t) ·XR(u; t) = |c(t)|2∫χR(x)e(u)dx+

∫χR(x)c · (x− c)e(u)dx, (7.21)

where the first term on the right is bounded from below by

E(u)−∫

(1− χR(x))e(u)dx

≥ ‖u(t)‖2L2x/2 +K1,0(u(t))− CER,0(t) ≥ δ0E(u)− CεE(u)& δ0E(u),

(7.22)

since ε ≪ δ0. The second term of (7.21) is dominated by splitting the integral into|x − c| ≤ R0 and |x − c| ≥ R0. In the interior it is bounded by using the energybound, and in the exterior it is bounded by using (7.6). Thus we obtain

∣∣∣∣∫χR(x)c · (x− c)e(u)dx

∣∣∣∣ . (R0 +Rε)E(u)|c|. (7.23)

In the same way we have

|XR(u; 0)|. (R0 +Rε)E(u), (7.24)

since c(0) = 0. Thus we get

δ0E(u)|c(t)|. (R0 +Rε+ εt)E(u), (7.25)

and sending t→ t0, we get

δ0R. εt0. (7.26)

�

7.3. Dispersion and contradiction. Finally we use the localized virial identity tosee dispersion of the critical element, which will contradict the above non-propagationproperty. For any R > 0, we define the localized virial VR(u; t) ∈ R by

VR(u; t) := 〈χR(x)ut|(x · ∇ +∇ · x)u〉, (7.27)

where χR is as defined in (1.23). Then we have for any solution u,

d

dtVR(u; t) = −

∫χR(x)[2|∇u|2 − d(D − 2)f(u)] +

d

2|u|2∆χR(x)dx

−∫r∂rχR(x)[|ut|2 + 2|ur|2 − |∇u|2 − |u|2 + 2f(u)]dx

≤ −Kd,−2(u(t)) + CER,0(u; t).

(7.28)

If u is a critical element, then u ∈ K+ and hence by Lemma 2.12, there existsδ2 ∈ (0, 1) such that

Kd,−2(u(t)) ≥ δ2‖∇u(t)‖2L2x

(7.29)


for all t > 0. Thus we obtain, integrating in t,

VR(u; t0) ≤ VR(u; 0)− δ2

∫ t0

0

‖∇u(t)‖2L2xdt+ CεE(u)t0. (7.30)

Now by the compactness Lemma 7.1, we have

Lemma 7.5. Let u be a forward critical element. Then for any ε > 0 there exists

C > 0 such that

‖u(t)‖2L2x≤ C‖∇u(t)‖2L2

x+ ε‖u(t)‖2L2

x, (7.31)

for all t > 0.

Proof. Otherwise there exists a sequence tn > 0 such that

‖u(tn)‖2L2x> n‖∇u(tn)‖2L2

x+ ε‖u(tn)‖2L2

x. (7.32)

Since u is L2x bounded, it follows ‖∇u(tn)‖L2

x→ 0. Then Lemma 7.1 implies that,

after passing to a subsequence, u(tn) → 0 strongly in H1x, then the above inequality

implies that u(tn) → 0 too. Hence EQ(u; tn) → 0, which contradicts the energyequivalence, Lemma 2.10. �

Multiplying the equation with u, and then applying the above lemma with ε =1/4, we obtain

∂t〈u|u〉 =∫

Rd

|u|2 − |∇u|2 − |u|2 +Df(u)dx.

≥∫

Rd

|u|2/2 + |u|2 − C|∇u|2dx,(7.33)

with some C > 0. Hence∫ t0

0

‖u‖2L2x+ ‖u‖2L2

xdt.E(u) +

∫ t0

0

‖∇u‖2L2xdt, (7.34)

and so

t0E(u) ≤∫ t0

0

EQ(u; t)dt.E(u) +

∫ t0

0

‖∇u‖2L2xdt. (7.35)

Now we choose positive ε ≪ δ2δ0 and R ≫ R0(ε). Then by Lemma 7.4 thereexists t0 ∼ δ0R/ε such that ER,0(u; t) ≤ εE(u) for 0 < t < t0. Then from (7.30) and(7.35), we have

−VR(u; t0) + VR(u; 0)& [δ2t0 − Cεt0 − C]E(u)& δ2t0E(u) ∼δ2δ0R

εE(u), (7.36)

while the left hand side is dominated by RE(u), which is a contradiction when ε/δ2δ0is sufficiently small. �


Appendix A. The range of scaling exponents

In Section 2, we have shown thatmα,β in (1.17) is positive and achieved (after mod-ification of the mass in the critical/exponential cases) if (α, β) satisfies (1.16). Herewe see that it is also necessary, modulo the obvious symmetry (α, β) → (−α,−β).For simplicity, we consider only the pure power nonlinearity.

Proposition A.1. Assume that neither (α, β) ∈ R2 nor (−α,−β) satisfies (1.16).Then there exists q ∈ (2⋆, 2

⋆) such that we have mα,β = −∞ for f(ϕ) = |ϕ|q.Proof. By symmetry with respect to (α, β) → (−α,−β), we may assume that β > 0and µ = 2α+ dβ > 0.

First we consider the case α < 0 and µ > 0, which implies that d ≥ 2. Let(2⋆, 2

⋆) ∋ q = 2 + p, then we have

αp+ µ ≥ dµ/(d− 2) > 0. (A.1)

Decompose K(ϕ) by

K = K1 +K2, K1(ϕ) = µ‖∇ϕ‖2L2

2, K2(ϕ) = µ

‖ϕ‖2L2

2− (αp+ µ)F (ϕ). (A.2)

Suppose that 0 6= ϕ ∈ H1(Rd) satisfies K2(ϕ) = 0. If there is no such ϕ, then K ispositive definite and the minimization set in (1.17) becomes empty. Let 1 < ν →1 + 0, then we have

0 > K2(νϕ) → K2(ϕ) = 0, K1(νϕ) → K1(ϕ) > 0. (A.3)

Now let λ(ν) > 0 solve

0 = K(νϕ(x/λ)) = λd−2K1(νϕ) + λdK2(νϕ), (A.4)

in other words λ(ν) = [−K2(νϕ)/K1(νϕ)]1/2. Then λ(ν) → ∞ as ν → 1 + 0 due to

(A.3). Since

µJ(ψ) = K(ψ) + β‖∇ψ‖2L2 + αpF (ψ), (A.5)

we obtain

µJ(νϕ(x/λ)) = βν2λd−2‖∇ϕ‖2L2 + αpλdF (νϕ) → −∞, (A.6)

which implies that m = −∞.Next, if µ = 0 > α, which implies d ≥ 2, then for any nonzero ϕ ∈ H1(Rd)

satisfying K(ϕ) = 0 we have

K(ϕ(x/λ)) = λdK(ϕ) = 0, (A.7)

and similarly as above, J(ϕ(x/λ)) = O(−λd) → ∞ as λ→ ∞.Finally consider the case µ < 0 < µ. Then αp + 2β = 0 has a solution p ∈

(4/d, 2⋆ − 2). Since αp+ µ = αp+ 2β + µ, there exists p ∈ (4/d, 2⋆ − 2) such that

αp+ µ < 0 < αp+ 2β. (A.8)


Then KN(ϕ) = −(αp + µ)F (ϕ) is positive and so for any ϕ ∈ H1(Rd), K(νϕ) ≥ 0if ν ≫ 1. Since the kinetic term in K is negative, there exists ξ(ν) ∈ Rd such thatK(eiξxνϕ) = 0. Since

−µJ(ψ) = −K(ψ) + 2β‖ϕ‖2L2

2− (αp+ 2β)F (ψ), (A.9)

we obtain

−µJ(eiξxνϕ) = 2βν2‖ϕ‖2L2

2− (αp+ 2β)F (νψ) → −∞, (A.10)

which implies that m = −∞. �

The above proof shows that if α < 0 and µ ≥ 0 then m = −∞ for all q ∈ (2, 2⋆].The choice of q was needed only in the other region.

Appendix B. Table of Notation

The notation below applies to any s ∈ R, ν ≥ 0, (α, β) ∈ R2, j, k ∈ Z, Z ∈ R

3,I ⊂ R, ϕ, ψ ∈ H1(Rd), u ∈ Ct(H

1x(R

d)), any suffix ♦,♥, any sequence ϕn ∈ H1(Rd),and any functional G on H1(Rd).

Dimension and scaling

d ∈ N, 2⋆, 2⋆ > 0: space dimension and critical powers (1.3)

α, β ∈ R, µ ≥ µ ≥ 0: scaling exponents and their functions (2.1)

ϕλα,β, Lα,βG: rescaled family and scaling derivative (1.13),(1.14)

(subscript of the form ♠α,β is often omitted as ♠)1st order representation

~u ↔ u: linked with each other by (4.1)Nonlinearity

F (ϕ), f(s) ≥ 0: nonlinear energy and its density (1.11)fS(s), fL(s) ≥ 0: small and large parts of f (1.24)p1, p2 > 0, κ0 ≥ 0: leading powers of fS and fL (1.25), (1.26), (1.29)

Functionals

J(ϕ), J (ν)(ϕ) ∈ R: static energy, with mass change (1.11), (1.12)

Kα,β(ϕ),K(c)α,β(ϕ) ∈ R, Hα,β(ϕ) ≥ 0: derivatives of J (1.15),(2.26)

KQα,β(ϕ),K

Nα,β(ϕ) ∈ R: quadratic and nonlinear parts of K (2.2)

E(u; t), E(ϕ,ψ), e(u) ∈ R: total energy and its density (1.5),(7.14)EQ(u; t), EQ(ϕ,ψ) ≥ 0: linear energy (1.37)

E(ϕ), Kα,β(ϕ) ∈ R: vector versions of E and K (4.3)P (u; t), ER,c(u; t) ∈ R: momentum and exterior energy (7.7),(7.5)XR(u; t), VR(u; t) ∈ R: localized energy center and virial (7.13),(7.27)

Variational splittings

mα,β, E⋆ ≥ 0: static and scattering energy thresholds (1.17),(6.1)

K±α,β, K+

α,β: splitting below the threshold (1.18),(4.5)

CνTM(G), C⋆TM(G) ∈ [0,∞]: Trudinger-Moser ratio (2.47),(2.49)

M(G) ∈ [0,∞]: Trudinger-Moser threshold on H1 (2.48)conc.G((ϕn)n) ∈ R: concentration at x = 0 (2.51)

Function spaces and exponents

[Z]ν(I), [Z]0(I), [Z]•ν(I): Lebesgue-Besov spaces on I × R

d (4.7)

Zs, Z∗(s) ∈ R3: regularity change and dual of exponents (4.9)


regθ(Z), strθ(Z),decθ(Z) ∈ R: regularity and decay indexes (4.8)H,W,K,M ♯, V ∈ R

3: exponents for d ∈ N (4.24),(4.31),(5.68)X,S,L ∈ R

3: exponents for d ≤ 4 (4.31),(4.54)

M,M, M , N ,N,Q, P, Y,R,G ∈ R3: exponents for d ≥ 5 (4.68),(4.69),(5.87)

Hε,Wε,M♯ε ∈ R3: exponents for d ≥ 5 (5.81)

H1ν , MC: H1(R2) and a set of Fourier multipliers on R

d (4.38), (5.2)

X , Y, Y0, Y, Y∗0 , Y∗: Strichartz-type spaces (4.54), (4.77)

ST (I), ST ∗(I), ST♦∞(I): Strichartz-type spaces on I × Rd (5.59),(5.60)

Profile decomposition

(t♦♥, x♦♥, h

♦♥) ∈ R

1+d × [0, 1]: time-space-scale shift parameter Section 5.1

γ♦♥ = −t♦♥/h♦♥ ∈ R: rescaled time shift

h♦∞ ∈ {0, 1},γ♦∞ ∈ [−∞,∞]: limit of h♦n and γ♦nT♦♥ϕ, 〈∇〉♦♥ϕ: operators dependent on (x♦♥, h

♦♥) (5.1)

(tjln , xjln , h

jln ), S

jln u: relative shift and transform (5.21),(5.20)

τ♦♥ ∈ R, τ♦∞ ∈ [−∞,∞]: scaled time shift and its limit (5.1)~U♦∞, U♦

∞, : nonlinear profiles (scaled limit) (5.52),(5.53)

~uj(n), ~u<k(n): nonlinear profiles (in original scales) (5.54),(5.58)

Acknowledgments

The authors thank Guixiang Xu for pointing out several mistakes in the firstmanuscript. S. Ibrahim is partially supported by NSERC# 371637-2009 grant anda start up fund from University of Victoria.

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Department of Mathematics and Statistics,, University of Victoria, PO Box 3060

STN CSC, Victoria, BC, V8P 5C3, Canada

E-mail address : [email protected]: http://www.math.uvic.ca/~ibrahim/

The Courant Institute for Mathematical Sciences,, New York University

E-mail address : [email protected]: http://www.math.nyu.edu/faculty/masmoudi

Department of Mathematics, Kyoto University

E-mail address : [email protected]



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Scattering threshold for the focusing nonlinear Klein–Gordon equation

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